# Complete Reachability Components, Reflexivity and Isolated-Vertex Partitions

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261028-r07-connected-component-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=0. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(0, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0001/)

Connected component · qf-mod28-component-of-vertex · candidate





## 2. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [2, 4], [2, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=1. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(1, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0002/)

Connected component · qf-mod28-component-of-vertex · candidate





## 3. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 4], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=2. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(2, [0, 1, 2, 3, 4, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0003/)

Connected component · qf-mod28-component-of-vertex · candidate





## 4. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=3. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(3, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0004/)

Connected component · qf-mod28-component-of-vertex · candidate





## 5. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 4], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=4. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(4, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0005/)

Connected component · qf-mod28-component-of-vertex · candidate





## 6. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=5. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(5, [0, 1, 2, 3, 4, 5])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0006/)

Connected component · qf-mod28-component-of-vertex · candidate





## 7. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [1, 3], [2, 3], [2, 4], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=6. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(6, [0, 1, 2, 3, 4, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0007/)

Connected component · qf-mod28-component-of-vertex · candidate





## 8. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 4], [2, 5], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=0. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(0, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0008/)

Connected component · qf-mod28-component-of-vertex · candidate





## 9. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 6], [1, 2], [1, 5], [1, 6], [2, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=1. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(1, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0009/)

Connected component · qf-mod28-component-of-vertex · candidate





## 10. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [2, 4], [2, 5], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=2. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(2, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0010/)

Connected component · qf-mod28-component-of-vertex · candidate





## 11. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [1, 2], [1, 6], [2, 4], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=3. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(3, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0011/)

Connected component · qf-mod28-component-of-vertex · candidate





## 12. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 5], [0, 6], [1, 4], [1, 5], [2, 4], [2, 5], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=4. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(4, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0012/)

Connected component · qf-mod28-component-of-vertex · candidate





## 13. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 6], [1, 5], [2, 4], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=5. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(5, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0013/)

Connected component · qf-mod28-component-of-vertex · candidate





## 14. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=6. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(6, [6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, [6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0014/)

Connected component · qf-mod28-component-of-vertex · candidate





## 15. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [2, 4], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=0. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(0, [0, 1])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0015/)

Connected component · qf-mod28-component-of-vertex · candidate





## 16. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 5], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 5], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=1. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(1, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0016/)

Connected component · qf-mod28-component-of-vertex · candidate





## 17. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=2. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(2, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0017/)

Connected component · qf-mod28-component-of-vertex · candidate





## 18. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=3. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(3, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0018/)

Connected component · qf-mod28-component-of-vertex · candidate





## 19. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=4. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(4, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0019/)

Connected component · qf-mod28-component-of-vertex · candidate





## 20. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 5], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=5. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(5, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0020/)

Connected component · qf-mod28-component-of-vertex · candidate





## 21. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=6. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(6, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0021/)

Connected component · qf-mod28-component-of-vertex · candidate





## 22. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [1, 3], [2, 3], [2, 5], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=0. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(0, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0022/)

Connected component · qf-mod28-component-of-vertex · candidate





## 23. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=1. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(1, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0023/)

Connected component · qf-mod28-component-of-vertex · candidate





## 24. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 6], [1, 2], [1, 5], [1, 6], [2, 5], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=2. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(2, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0024/)

Connected component · qf-mod28-component-of-vertex · candidate





## 25. Return the Entire Component, Not a Traversal Order

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [3, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=3. Give (vertex,component), with its vertices in increasing label order.

View answer and explanation(3, [0, 1, 2, 3, 4, 5, 6])

Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0025/)

Connected component · qf-mod28-component-of-vertex · candidate





## 26. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0026/)

Connected component · qf-mod28-component-all · candidate





## 27. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [2, 4], [2, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0027/)

Connected component · qf-mod28-component-all · candidate





## 28. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 4], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 6], [5]], 2)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 6], [5]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0028/)

Connected component · qf-mod28-component-all · candidate





## 29. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0029/)

Connected component · qf-mod28-component-all · candidate





## 30. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 4], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0030/)

Connected component · qf-mod28-component-all · candidate





## 31. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5], [6]], 2)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0031/)

Connected component · qf-mod28-component-all · candidate





## 32. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [1, 3], [2, 3], [2, 4], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 6], [5]], 2)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 6], [5]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0032/)

Connected component · qf-mod28-component-all · candidate





## 33. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 4], [2, 5], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0033/)

Connected component · qf-mod28-component-all · candidate





## 34. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 6], [1, 2], [1, 5], [1, 6], [2, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0034/)

Connected component · qf-mod28-component-all · candidate





## 35. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [2, 4], [2, 5], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0035/)

Connected component · qf-mod28-component-all · candidate





## 36. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [1, 2], [1, 6], [2, 4], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0036/)

Connected component · qf-mod28-component-all · candidate





## 37. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 5], [0, 6], [1, 4], [1, 5], [2, 4], [2, 5], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0037/)

Connected component · qf-mod28-component-all · candidate





## 38. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 6], [1, 5], [2, 4], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0038/)

Connected component · qf-mod28-component-all · candidate





## 39. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5], [6]], 2)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0039/)

Connected component · qf-mod28-component-all · candidate





## 40. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [2, 4], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1], [2, 3, 4, 5, 6]], 2)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5, 6]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0040/)

Connected component · qf-mod28-component-all · candidate





## 41. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 5], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 5], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0041/)

Connected component · qf-mod28-component-all · candidate





## 42. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0042/)

Connected component · qf-mod28-component-all · candidate





## 43. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0043/)

Connected component · qf-mod28-component-all · candidate





## 44. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0044/)

Connected component · qf-mod28-component-all · candidate





## 45. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 5], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0045/)

Connected component · qf-mod28-component-all · candidate





## 46. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0046/)

Connected component · qf-mod28-component-all · candidate





## 47. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [1, 3], [2, 3], [2, 5], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0047/)

Connected component · qf-mod28-component-all · candidate





## 48. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0048/)

Connected component · qf-mod28-component-all · candidate





## 49. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 6], [1, 2], [1, 5], [1, 6], [2, 5], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0049/)

Connected component · qf-mod28-component-all · candidate





## 50. Partition the Full Vertex Carrier into Reachability Classes

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [3, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], 1)

Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0050/)

Connected component · qf-mod28-component-all · candidate





## 51. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=0, right=1. Give (left,right,left_component,same_component?).

View answer and explanation(0, 1, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, 1, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0051/)

Connected component · qf-mod28-component-same-audit · candidate





## 52. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [2, 4], [2, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=1, right=4. Give (left,right,left_component,same_component?).

View answer and explanation(1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, 4, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0052/)

Connected component · qf-mod28-component-same-audit · candidate





## 53. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 4], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=2, right=0. Give (left,right,left_component,same_component?).

View answer and explanation(2, 0, [0, 1, 2, 3, 4, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 1, 2, 3, 4, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0053/)

Connected component · qf-mod28-component-same-audit · candidate





## 54. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=3, right=3. Give (left,right,left_component,same_component?).

View answer and explanation(3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, 3, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0054/)

Connected component · qf-mod28-component-same-audit · candidate





## 55. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 4], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=4, right=6. Give (left,right,left_component,same_component?).

View answer and explanation(4, 6, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, 6, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0055/)

Connected component · qf-mod28-component-same-audit · candidate





## 56. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=5, right=2. Give (left,right,left_component,same_component?).

View answer and explanation(5, 2, [0, 1, 2, 3, 4, 5], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, 2, [0, 1, 2, 3, 4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0056/)

Connected component · qf-mod28-component-same-audit · candidate





## 57. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [1, 3], [2, 3], [2, 4], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=6, right=5. Give (left,right,left_component,same_component?).

View answer and explanation(6, 5, [0, 1, 2, 3, 4, 6], No)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, 5, [0, 1, 2, 3, 4, 6], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0057/)

Connected component · qf-mod28-component-same-audit · candidate





## 58. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 4], [2, 5], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=0, right=1. Give (left,right,left_component,same_component?).

View answer and explanation(0, 1, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, 1, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0058/)

Connected component · qf-mod28-component-same-audit · candidate





## 59. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 6], [1, 2], [1, 5], [1, 6], [2, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=1, right=4. Give (left,right,left_component,same_component?).

View answer and explanation(1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, 4, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0059/)

Connected component · qf-mod28-component-same-audit · candidate





## 60. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [2, 4], [2, 5], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=2, right=0. Give (left,right,left_component,same_component?).

View answer and explanation(2, 0, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0060/)

Connected component · qf-mod28-component-same-audit · candidate





## 61. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [1, 2], [1, 6], [2, 4], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=3, right=3. Give (left,right,left_component,same_component?).

View answer and explanation(3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, 3, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0061/)

Connected component · qf-mod28-component-same-audit · candidate





## 62. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 5], [0, 6], [1, 4], [1, 5], [2, 4], [2, 5], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=4, right=6. Give (left,right,left_component,same_component?).

View answer and explanation(4, 6, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, 6, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0062/)

Connected component · qf-mod28-component-same-audit · candidate





## 63. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 6], [1, 5], [2, 4], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=5, right=2. Give (left,right,left_component,same_component?).

View answer and explanation(5, 2, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, 2, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0063/)

Connected component · qf-mod28-component-same-audit · candidate





## 64. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=6, right=5. Give (left,right,left_component,same_component?).

View answer and explanation(6, 5, [6], No)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, 5, [6], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0064/)

Connected component · qf-mod28-component-same-audit · candidate





## 65. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [2, 4], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=0, right=1. Give (left,right,left_component,same_component?).

View answer and explanation(0, 1, [0, 1], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, 1, [0, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0065/)

Connected component · qf-mod28-component-same-audit · candidate





## 66. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 5], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 5], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=1, right=4. Give (left,right,left_component,same_component?).

View answer and explanation(1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, 4, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0066/)

Connected component · qf-mod28-component-same-audit · candidate





## 67. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=2, right=0. Give (left,right,left_component,same_component?).

View answer and explanation(2, 0, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0067/)

Connected component · qf-mod28-component-same-audit · candidate





## 68. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=3, right=3. Give (left,right,left_component,same_component?).

View answer and explanation(3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, 3, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0068/)

Connected component · qf-mod28-component-same-audit · candidate





## 69. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=4, right=6. Give (left,right,left_component,same_component?).

View answer and explanation(4, 6, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, 6, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0069/)

Connected component · qf-mod28-component-same-audit · candidate





## 70. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 5], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=5, right=2. Give (left,right,left_component,same_component?).

View answer and explanation(5, 2, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, 2, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0070/)

Connected component · qf-mod28-component-same-audit · candidate





## 71. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=6, right=5. Give (left,right,left_component,same_component?).

View answer and explanation(6, 5, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, 5, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0071/)

Connected component · qf-mod28-component-same-audit · candidate





## 72. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [1, 3], [2, 3], [2, 5], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=0, right=1. Give (left,right,left_component,same_component?).

View answer and explanation(0, 1, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, 1, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0072/)

Connected component · qf-mod28-component-same-audit · candidate





## 73. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=1, right=4. Give (left,right,left_component,same_component?).

View answer and explanation(1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, 4, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0073/)

Connected component · qf-mod28-component-same-audit · candidate





## 74. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 6], [1, 2], [1, 5], [1, 6], [2, 5], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=2, right=0. Give (left,right,left_component,same_component?).

View answer and explanation(2, 0, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0074/)

Connected component · qf-mod28-component-same-audit · candidate





## 75. Reflexive Connectivity Includes a Length-Zero Path

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [3, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=3, right=3. Give (left,right,left_component,same_component?).

View answer and explanation(3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)

Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, 3, [0, 1, 2, 3, 4, 5, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0075/)

Connected component · qf-mod28-component-same-audit · candidate





## 76. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0076/)

Connected component · qf-mod28-component-partition · candidate





## 77. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [2, 4], [2, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0077/)

Connected component · qf-mod28-component-partition · candidate





## 78. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 4], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 6], [5]], Yes, Yes, [[5]], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 6], [5]], Yes, Yes, [[5]], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0078/)

Connected component · qf-mod28-component-partition · candidate





## 79. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0079/)

Connected component · qf-mod28-component-partition · candidate





## 80. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 4], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0080/)

Connected component · qf-mod28-component-partition · candidate





## 81. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5], [6]], Yes, Yes, [[6]], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], Yes, Yes, [[6]], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0081/)

Connected component · qf-mod28-component-partition · candidate





## 82. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [1, 3], [2, 3], [2, 4], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 6], [5]], Yes, Yes, [[5]], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 6], [5]], Yes, Yes, [[5]], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0082/)

Connected component · qf-mod28-component-partition · candidate





## 83. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 4], [2, 5], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0083/)

Connected component · qf-mod28-component-partition · candidate





## 84. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 6], [1, 2], [1, 5], [1, 6], [2, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0084/)

Connected component · qf-mod28-component-partition · candidate





## 85. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [2, 4], [2, 5], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0085/)

Connected component · qf-mod28-component-partition · candidate





## 86. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [1, 2], [1, 6], [2, 4], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0086/)

Connected component · qf-mod28-component-partition · candidate





## 87. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 5], [0, 6], [1, 4], [1, 5], [2, 4], [2, 5], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0087/)

Connected component · qf-mod28-component-partition · candidate





## 88. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 6], [1, 5], [2, 4], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0088/)

Connected component · qf-mod28-component-partition · candidate





## 89. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5], [6]], Yes, Yes, [[6]], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], Yes, Yes, [[6]], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0089/)

Connected component · qf-mod28-component-partition · candidate





## 90. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [2, 4], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1], [2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0090/)

Connected component · qf-mod28-component-partition · candidate





## 91. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 5], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 5], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0091/)

Connected component · qf-mod28-component-partition · candidate





## 92. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0092/)

Connected component · qf-mod28-component-partition · candidate





## 93. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0093/)

Connected component · qf-mod28-component-partition · candidate





## 94. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0094/)

Connected component · qf-mod28-component-partition · candidate





## 95. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 5], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0095/)

Connected component · qf-mod28-component-partition · candidate





## 96. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0096/)

Connected component · qf-mod28-component-partition · candidate





## 97. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [1, 3], [2, 3], [2, 5], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0097/)

Connected component · qf-mod28-component-partition · candidate





## 98. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0098/)

Connected component · qf-mod28-component-partition · candidate





## 99. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 6], [1, 2], [1, 5], [1, 6], [2, 5], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0099/)

Connected component · qf-mod28-component-partition · candidate





## 100. Coverage and Disjointness Must Include Isolated Vertices

Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [3, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.

View answer and explanation([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)

Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261028-r07-connected-component-0100/)

Connected component · qf-mod28-component-partition · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261028-r07-connected-component-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261028-r07-connected-component-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261028-r07-connected-component-foundations/validation-report.json)