Original question and givens
Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 6, 7], 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(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 6, 7], 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).
Step-by-step derivation
S001 · implies
Before
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 dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
complete original vertex carrier Fin(5);canonical original edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]];raw selected edge indices=[0, 3, 6, 7]
After
sorted unique ORIGINAL selection=[0, 3, 6, 7];selected edges=[[0, 1], [1, 2], [2, 3], [3, 4]];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 dependencies
Rule: spantree_original_graph_whole_carrier_not_matching_domain · Uses: G001, S001
S003 · implies
Before
parent graph:actual edge index=0;edge=[0, 1];reached from=0;new vertex=1
After
original-edge path from0=[0, 1];reached vertices=[0, 1]
Conditions: none
Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts.
Step identifiers and dependencies
Rule: spantree_every_discovered_vertex_path_witness · Uses: G001, S002
S004 · implies
Before
parent graph:actual edge index=1;edge=[0, 2];reached from=0;new vertex=2
After
original-edge path from0=[0, 2];reached vertices=[0, 1, 2]
Conditions: none
Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts.
Step identifiers and dependencies
Rule: spantree_every_discovered_vertex_path_witness · Uses: G001, S003
S005 · implies
Before
parent graph:actual edge index=2;edge=[0, 3];reached from=0;new vertex=3
After
original-edge path from0=[0, 3];reached vertices=[0, 1, 2, 3]
Conditions: none
Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts.
Step identifiers and dependencies
Rule: spantree_every_discovered_vertex_path_witness · Uses: G001, S004
S006 · implies
Before
parent graph:actual edge index=5;edge=[1, 4];reached from=1;new vertex=4
After
original-edge path from0=[0, 1, 4];reached vertices=[0, 1, 2, 3, 4]
Conditions: none
Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts.
Step identifiers and dependencies
Rule: spantree_every_discovered_vertex_path_witness · Uses: G001, S005
S007 · implies
Before
parent graph:complete carrier Fin(5);ALL graph edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]];reached=[0, 1, 2, 3, 4];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 dependencies
Rule: spantree_reachability_and_closed_cut_completeness · Uses: G001, S006
S008 · implies
Before
selected spanning subgraph:actual edge index=0;edge=[0, 1];reached from=0;new vertex=1
After
original-edge path from0=[0, 1];reached vertices=[0, 1]
Conditions: none
Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts.
Step identifiers and dependencies
Rule: spantree_every_discovered_vertex_path_witness · Uses: G001, S007
S009 · implies
Before
selected spanning subgraph:actual edge index=1;edge=[1, 2];reached from=1;new vertex=2
After
original-edge path from0=[0, 1, 2];reached vertices=[0, 1, 2]
Conditions: none
Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts.
Step identifiers and dependencies
Rule: spantree_every_discovered_vertex_path_witness · Uses: G001, S008
S010 · implies
Before
selected spanning subgraph:actual edge index=2;edge=[2, 3];reached from=2;new vertex=3
After
original-edge path from0=[0, 1, 2, 3];reached vertices=[0, 1, 2, 3]
Conditions: none
Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts.
Step identifiers and dependencies
Rule: spantree_every_discovered_vertex_path_witness · Uses: G001, S009
S011 · implies
Before
selected spanning subgraph:actual edge index=3;edge=[3, 4];reached from=3;new vertex=4
After
original-edge path from0=[0, 1, 2, 3, 4];reached vertices=[0, 1, 2, 3, 4]
Conditions: none
Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts.
Step identifiers and dependencies
Rule: spantree_every_discovered_vertex_path_witness · Uses: G001, S010
S012 · implies
Before
selected spanning subgraph:complete carrier Fin(5);ALL graph edges=[[0, 1], [1, 2], [2, 3], [3, 4]];reached=[0, 1, 2, 3, 4];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 dependencies
Rule: spantree_reachability_and_closed_cut_completeness · Uses: G001, S011
S013 · implies
Before
original selected edge index=0;edge=[0, 1];previous forest component roots=[0, 1]
After
closes cycle=No;actual cycle witness=[]
Conditions: none
If endpoints were already in the same forest component,the previous forest path plus this actual edge gives a cycle.Otherwise joining two components preserves acyclicity.Check every selected edge,not merely n−1 counts.
Step identifiers and dependencies
Rule: spantree_every_edge_cycle_or_forest_extension · Uses: G001, S012
S014 · implies
Before
original selected edge index=3;edge=[1, 2];previous forest component roots=[0, 2]
After
closes cycle=No;actual cycle witness=[]
Conditions: none
If endpoints were already in the same forest component,the previous forest path plus this actual edge gives a cycle.Otherwise joining two components preserves acyclicity.Check every selected edge,not merely n−1 counts.
Step identifiers and dependencies
Rule: spantree_every_edge_cycle_or_forest_extension · Uses: G001, S013
S015 · implies
Before
original selected edge index=6;edge=[2, 3];previous forest component roots=[0, 3]
After
closes cycle=No;actual cycle witness=[]
Conditions: none
If endpoints were already in the same forest component,the previous forest path plus this actual edge gives a cycle.Otherwise joining two components preserves acyclicity.Check every selected edge,not merely n−1 counts.
Step identifiers and dependencies
Rule: spantree_every_edge_cycle_or_forest_extension · Uses: G001, S014
S016 · implies
Before
original selected edge index=7;edge=[3, 4];previous forest component roots=[0, 4]
After
closes cycle=No;actual cycle witness=[]
Conditions: none
If endpoints were already in the same forest component,the previous forest path plus this actual edge gives a cycle.Otherwise joining two components preserves acyclicity.Check every selected edge,not merely n−1 counts.
Step identifiers and dependencies
Rule: spantree_every_edge_cycle_or_forest_extension · Uses: G001, S015
S017 · implies
Before
same complete vertex carrier Fin(5);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 dependencies
Rule: spantree_connectivity_acyclicity_full_vertices_conjunction · Uses: G001, S016
S018 · implies
Before
ACTUAL carrier size=5;ACTUAL selected indices=[0, 3, 6, 7];independent tree validity=Yes
After
actual one-vertex empty-selection condition=No
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 dependencies
Rule: spantree_one_vertex_flag_not_forced_input_type · Uses: G001, S017