# Question 8447: Components, Cycles, and Nonempty Tree Qualification

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0015-08447 · q-20261012-advance-20261017-r06-tree-0022



## Original question and givens

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.


- G001 — Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-tree](https://ocme.evemisslab.com/en/mko/mko-tree/)





## Step-by-step derivation

S001 · impliesBefore

n=5;raw edges=[[1, 2]]

After

canonical edges=[[1, 2]]

Conditions: none

Validate every simple undirected edge for typed distinct endpoints.Sort endpoints and count reversed or repeated edges once;do not infer adjacency from a drawing.

Step identifiers and dependenciesRule: simple_graph_edge_normalization · Uses: G001





S002 · impliesBefore

phase=original;vertex=0;all edges=[[1, 2]]

After

neighbors=[];degree=0

Conditions: none

Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels.

Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S001





S003 · impliesBefore

phase=original;vertex=1;all edges=[[1, 2]]

After

neighbors=[2];degree=1

Conditions: none

Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels.

Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S002





S004 · impliesBefore

phase=original;vertex=2;all edges=[[1, 2]]

After

neighbors=[1];degree=1

Conditions: none

Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels.

Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S003





S005 · impliesBefore

phase=original;vertex=3;all edges=[[1, 2]]

After

neighbors=[];degree=0

Conditions: none

Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels.

Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S004





S006 · impliesBefore

phase=original;vertex=4;all edges=[[1, 2]]

After

neighbors=[];degree=0

Conditions: none

Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels.

Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S005





S007 · impliesBefore

phase=original;root=0;depth=0;frontier=[0]

After

next newly reached=[];component seen=[0]

Conditions: none

Expand along canonical edges only to unseen original labels.A component is complete only when its frontier empties;then start at another unseen vertex.

Step identifiers and dependenciesRule: tree_complete_reachability_layer · Uses: G001, S006





S008 · impliesBefore

phase=original;root=1;depth=0;frontier=[1]

After

next newly reached=[2];component seen=[1, 2]

Conditions: none

Expand along canonical edges only to unseen original labels.A component is complete only when its frontier empties;then start at another unseen vertex.

Step identifiers and dependenciesRule: tree_complete_reachability_layer · Uses: G001, S007





S009 · impliesBefore

phase=original;root=1;depth=1;frontier=[2]

After

next newly reached=[];component seen=[1, 2]

Conditions: none

Expand along canonical edges only to unseen original labels.A component is complete only when its frontier empties;then start at another unseen vertex.

Step identifiers and dependenciesRule: tree_complete_reachability_layer · Uses: G001, S008





S010 · impliesBefore

phase=original;root=3;depth=0;frontier=[3]

After

next newly reached=[];component seen=[3]

Conditions: none

Expand along canonical edges only to unseen original labels.A component is complete only when its frontier empties;then start at another unseen vertex.

Step identifiers and dependenciesRule: tree_complete_reachability_layer · Uses: G001, S009





S011 · impliesBefore

phase=original;root=4;depth=0;frontier=[4]

After

next newly reached=[];component seen=[4]

Conditions: none

Expand along canonical edges only to unseen original labels.A component is complete only when its frontier empties;then start at another unseen vertex.

Step identifiers and dependenciesRule: tree_complete_reachability_layer · Uses: G001, S010





S012 · impliesBefore

phase=original;spanning forest=[[1, 2]];all edges=[[1, 2]]

After

nonforest edges=[];cyclic=No

Conditions: none

First-discovery edges form a tree in each reached component.Any remaining edge closes an existing tree path into a cycle.Components were established first;this is not an edge-count-only tree test.

Step identifiers and dependenciesRule: spanning_forest_cycle_witness · Uses: G001, S011





S013 · impliesBefore

phase=original;vertices=[0, 1, 2, 3, 4];components=[[0], [1, 2], [3], [4]];cyclic=No

After

pairwise reachable=No;nonempty=Yes;tree=No

Conditions: none

A tree must be nonempty,have one component and be acyclic.Pairwise reachability is vacuously true on an empty vertex set,but nonemptiness fails.A singleton with no edges is a tree;its degree0 vertex is not a leaf.

Step identifiers and dependenciesRule: nonempty_connected_acyclic_tree_gate · Uses: G001, S012









## Reverse checks


- S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(5, [[1, 2]], [[0], [1, 2], [3], [4]], No, No, Yes, No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 865487c3e3dbcdfc8002222cd01a1d16a06f24eb55d433b0d89b0265924fa167
- candidate record SHA-256: 27e6837468ab50db511026b5307d1b768d40193da017159e425de5352c71de13
- semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18
- question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r06-tree-0022.json)