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.
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.
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.
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.
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.
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.
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.
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.