Complete Reachability Components, Reflexivity and Isolated-Vertex Partitions
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Connected component · qf-mod28-component-partition · candidate
Canonical question JSONL · English JSONL · Validation report