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Question 8095: Induced Graph with Original Vertex Labels

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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R0015-08095 · q-20261012-advance-20261017-r05-graph-0095

Original question and givens

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. 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. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).

  • G001 — Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. 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. Retain source subset=[4, 2, 0], normalized as a set. Give (old normalized edges, sorted retained labels, induced edges, retained-label degrees, retained-label neighbor lists).

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=[[0, 1], [0, 2], [0, 3], [0, 4]]

After

canonical edges=[[0, 1], [0, 2], [0, 3], [0, 4]]

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=[[0, 1], [0, 2], [0, 3], [0, 4]]

After

neighbors=[1, 2, 3, 4];degree=4

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=[[0, 1], [0, 2], [0, 3], [0, 4]]

After

neighbors=[0];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=[[0, 1], [0, 2], [0, 3], [0, 4]]

After

neighbors=[0];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=[[0, 1], [0, 2], [0, 3], [0, 4]]

After

neighbors=[0];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, S004

S006 · implies

Before

phase=original;vertex=4;all edges=[[0, 1], [0, 2], [0, 3], [0, 4]]

After

neighbors=[0];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, S005

S007 · implies

Before

raw subset=[4, 2, 0];old edges=[[0, 1], [0, 2], [0, 3], [0, 4]]

After

retained original labels=[0, 2, 4];induced edges=[[0, 2], [0, 4]]

Conditions: none

Deduplicate subset labels and retain exactly original edges with both endpoints present.Do not renumber nonconsecutive labels.

Step identifiers and dependencies

Rule: induced_graph_both_endpoints_guard · Uses: G001, S006

S008 · implies

Before

phase=induced;vertex=0;all edges=[[0, 2], [0, 4]]

After

neighbors=[2, 4];degree=2

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, S007

S009 · implies

Before

phase=induced;vertex=2;all edges=[[0, 2], [0, 4]]

After

neighbors=[0];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, S008

S010 · implies

Before

phase=induced;vertex=4;all edges=[[0, 2], [0, 4]]

After

neighbors=[0];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, S009

Reverse checks

  • 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

([[0, 1], [0, 2], [0, 3], [0, 4]], [0, 2, 4], [[0, 2], [0, 4]], [2, 1, 1], [[2, 4], [0], [0]])

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: 24afddc902c7ed9900db08e5850092edf6c3515c2b8dc848ec2f846799e0858d
  • candidate record SHA-256: 7cb8d3b9d154fbcfa81d6bb4c600964d07e35d8b428e1eace03315e70f6e5626
  • semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18
  • question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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