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Question 8447: Components, Cycles, and Nonempty Tree Qualification

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: simple_graph_edge_normalization · Uses: G001

S002 · implies

Before

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 dependencies

Rule: complete_graph_neighbor_inventory · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: complete_graph_neighbor_inventory · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: complete_graph_neighbor_inventory · Uses: G001, S003

S005 · implies

Before

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 dependencies

Rule: complete_graph_neighbor_inventory · Uses: G001, S004

S006 · implies

Before

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 dependencies

Rule: complete_graph_neighbor_inventory · Uses: G001, S005

S007 · implies

Before

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 dependencies

Rule: tree_complete_reachability_layer · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: tree_complete_reachability_layer · Uses: G001, S007

S009 · implies

Before

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 dependencies

Rule: tree_complete_reachability_layer · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: tree_complete_reachability_layer · Uses: G001, S009

S011 · implies

Before

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 dependencies

Rule: tree_complete_reachability_layer · Uses: G001, S010

S012 · implies

Before

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 dependencies

Rule: spanning_forest_cycle_witness · Uses: G001, S011

S013 · implies

Before

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 dependencies

Rule: 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
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