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Question 8463: Degree-One Leaf Deletion without Renumbering

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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R0015-08463 · q-20261012-advance-20261017-r06-tree-0063

Original question and givens

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. 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. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

  • G001 — Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. 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. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

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=6;raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]

After

canonical edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]

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, 2], [0, 3], [0, 4], [0, 5], [1, 5]]

After

neighbors=[2, 3, 4, 5];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, 2], [0, 3], [0, 4], [0, 5], [1, 5]]

After

neighbors=[5];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, 2], [0, 3], [0, 4], [0, 5], [1, 5]]

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, 2], [0, 3], [0, 4], [0, 5], [1, 5]]

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, 2], [0, 3], [0, 4], [0, 5], [1, 5]]

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

phase=original;vertex=5;all edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]

After

neighbors=[0, 1];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, S006

S008 · implies

Before

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

After

next newly reached=[2, 3, 4, 5];component seen=[0, 2, 3, 4, 5]

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=0;depth=1;frontier=[2, 3, 4, 5]

After

next newly reached=[1];component seen=[0, 1, 2, 3, 4, 5]

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=0;depth=2;frontier=[1]

After

next newly reached=[];component seen=[0, 1, 2, 3, 4, 5]

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;spanning forest=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]];all edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]

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, S010

S012 · implies

Before

phase=original;vertices=[0, 1, 2, 3, 4, 5];components=[[0, 1, 2, 3, 4, 5]];cyclic=No

After

pairwise reachable=Yes;nonempty=Yes;tree=Yes

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, S011

S013 · implies

Before

original tree=Yes;at=5;typed=Yes;degree/status=2

After

qualified degree-one leaf=No

Conditions: none

Require an original tree,a typed vertex and original degree exactly1.The degree0 vertex of a singleton tree is not a leaf.

Step identifiers and dependencies

Rule: original_tree_degree_one_leaf_guard · 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

(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)

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: bc9e6d03169f21f7697c1677d2689359ba4736b772831a432e8acd93c40d7cda
  • candidate record SHA-256: 24ca318751be113bdbf95b5fd65af66bd5c69983f5ad2a0c18331879ed7df354
  • semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18
  • question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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