OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 35456: Nonempty Finite-Graph Tree Eligibility

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

Return to catalogue page 178

← Question derivation index · Original question batch

R0069-35456 · q-20261007-advance-20261009-r02-tree-0006

Original question and givens

The finite simple undirected graph has V=[0, 1, 2] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

  • G001 — The finite simple undirected graph has V=[0, 1, 2] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

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

source roles = [qf-tree-property, [E, V, domain, operation]]

After

derived = [tree_property, finite_simple_undirected]

Conditions: none

Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.

Step identifiers and dependencies

Rule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001

S002 · implies

Before

source roles = [TREE vertices ORIGINAL, 10, [0, 1, 2]]

After

derived = [[0, 1, 2]]

Conditions: none

Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits.

Step identifiers and dependencies

Rule: Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort · Uses: G001, S001

S003 · implies

Before

source roles = [TREE raw edges ORIGINAL, 0, [0, 1]]

After

derived = [[0, 1]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

Step identifiers and dependencies

Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S002

S004 · implies

Before

source roles = [[0, 1, 2], [[0, 1]], Yes]

After

derived = [[[0, 1]], [[0, [1]], [1, [0]], [2, []]]]

Conditions: none

Type the WHOLE raw edge list,then normalize undirected endpoints/deduplicate;reversed duplicates are allowed.INPUT edges have no loops/outside endpoints,while isolated vertices stay in V.TREE raw E does NOT acquire LDL’s200-entry limit.

Step identifiers and dependencies

Rule: Graph_WHOLE_simple_undirected_edges_normalized_NO_loops_or_outside · Uses: G001, S003

S005 · implies

Before

source roles = [0, 1, 0, 1]

After

derived = [Yes]

Conditions: none

Merge endpoint components ONLY when different.If already in the same component,this edge closes a cycle.Check EVERY edge across ALL components,not only one reached block.

Step identifiers and dependencies

Rule: Tree_EACH_edge_union_components_cycle_audit · Uses: G001, S004

S006 · implies

Before

source roles = [0, [1]]

After

derived = [[0, 1], [0]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependencies

Rule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S005

S007 · implies

Before

source roles = [1, [0]]

After

derived = [[0, 1], [0, 1]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependencies

Rule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S006

S008 · implies

Before

source roles = [[0, 1, 2], [[0, 1]]]

After

derived = [Yes, No, Yes, No]

Conditions: none

THIS entry defines a tree as NONEMPTY,connected and acyclic.Empty graph is No;an edgeless singleton is Yes.A triangle plus an isolated vertex is No EVEN when|E|=|V|−1.

Step identifiers and dependencies

Rule: Tree_NONEMPTY_connected_acyclic_ALL_three · Uses: G001, S007

Reverse checks

  • S008 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S007 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S006 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S005 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S004 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S003 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S002 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S001 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.

Result

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: 0dc4151ad6f640af7cbf1c2d7507c2413ae2ad8f9e89ecd07790cd09eee04be9
  • candidate record SHA-256: 6e402ce6eb92da752759a6d297fb1ead1752ad61485677b8dda6927483173443
  • semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b
  • question-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON