# Question 18476: Recognize the Single-Vertex Empty-Edge Tree without Assuming It [Return to catalogue page 93](https://ocme.evemisslab.com/en/questions/page/93/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261104-r06-spanning-tree-foundations/#q-20261014-advance-20261104-r06-spanning-tree-0051) 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. R0035-18476 · q-20261014-advance-20261104-r06-spanning-tree-0051 ## Original question and givens Use the complete finite undirected loop-free simple parent graph on Fin(1)={0,…,0}, canonical sorted edges=[], and distinct parent-edge selection=[], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection). - G001 — Use the complete finite undirected loop-free simple parent graph on Fin(1)={0,…,0}, canonical sorted edges=[], and distinct parent-edge selection=[], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection). ## 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-spanning-tree](https://ocme.evemisslab.com/en/mko/mko-spanning-tree/) ## Step-by-step derivation S001 · impliesBefore source operation=spantree-one-vertex;original snapshot and mathematics retained After explicit ordered source output fields=[vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection] Conditions: none Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore complete original vertex carrier Fin(1);canonical original edges=[];raw selected edge indices=[] After sorted unique ORIGINAL selection=[];selected edges=[];ALL parent vertices retained;no10-edge cap or weights Conditions: none This family bounds vertices by12 but has no maximum-matching10-edge cap.Select only original edges,retain every vertex including isolates,and normalize selection order without changing original indices.No weights or minimization objective are supplied. Step identifiers and dependenciesRule: spantree_original_graph_whole_carrier_not_matching_domain · Uses: G001, S001 S003 · impliesBefore parent graph:complete carrier Fin(1);ALL graph edges=[];reached=[0];unreached=[] After every graph edge stays within reached or its complement;connected=Yes Conditions: none After the queue is exhausted,no edge crosses the reached/unreached cut.Unreached vertices therefore have no path from0;otherwise path witnesses cover the carrier.A one-vertex empty-edge graph is connected via its initial vertex. Step identifiers and dependenciesRule: spantree_reachability_and_closed_cut_completeness · Uses: G001, S002 S004 · impliesBefore selected spanning subgraph:complete carrier Fin(1);ALL graph edges=[];reached=[0];unreached=[] After every graph edge stays within reached or its complement;connected=Yes Conditions: none After the queue is exhausted,no edge crosses the reached/unreached cut.Unreached vertices therefore have no path from0;otherwise path witnesses cover the carrier.A one-vertex empty-edge graph is connected via its initial vertex. Step identifiers and dependenciesRule: spantree_reachability_and_closed_cut_completeness · Uses: G001, S003 S005 · impliesBefore same complete vertex carrier Fin(1);parent connected=Yes;selected connected=Yes;all cycle witnesses=[] After selected acyclic=Yes;spanning tree validity=Yes Conditions: none A disconnected parent has no spanning tree,but failure of one selection in a connected parent does not refute existence of another.This audit retains all vertices and conjoins selected connectivity and acyclicity without dropping isolates. Step identifiers and dependenciesRule: spantree_connectivity_acyclicity_full_vertices_conjunction · Uses: G001, S004 S006 · impliesBefore ACTUAL carrier size=1;ACTUAL selected indices=[];independent tree validity=Yes After actual one-vertex empty-selection condition=Yes Conditions: none The final flag tests the actual one-vertex empty-selection case.For n>1 it may be false while tree validity is true;the family name does not force n=1. Step identifiers and dependenciesRule: spantree_one_vertex_flag_not_forced_input_type · Uses: G001, S005 ## Reverse checks - 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 (1, [], Yes, Yes) 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: c93712cafd793bf24b41337f83a522777b97ea1358984ddb6edeb2acbfbb2bf3 - candidate record SHA-256: c653b5b49ff04420b63b153c2a2f01468dd7756c8b6ecabd35da49713d851567 - semantic review receipt SHA-256: 713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc - question-first witness SHA-256: 0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261104-r06-spanning-tree-0051.json)