S001 · implies
Before
full carrier Fin(7);raw undirected edges=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]
After
canonical simple edges=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]];full adjacency rows=[[2, 5], [4], [0, 4, 5], [4, 5], [1, 2, 3], [0, 2, 3], []]
Conditions: none
Build adjacency on the full carrier,rejecting loops and duplicate undirected edges.Vertices absent from edges retain empty rows;do not switch to directed connectivity.
Step identifiers and dependencies
Rule: component_full_undirected_graph_includes_isolates · Uses: G001
S002 · implies
Before
selected vertex:start vertex=6;complete adjacency=[[2, 5], [4], [0, 4, 5], [4, 5], [1, 2, 3], [0, 2, 3], []]
After
initial visited=[6];path to start=[6];edge length0
Conditions: none
The start reaches itself by a length-zero path.Even an isolated vertex must be included rather than omitted for having no edges.
Step identifiers and dependencies
Rule: component_zero_length_reflexive_path · Uses: G001, S001
S003 · implies
Before
selected vertex:layer=0;frontier=[6];all frontier neighbors=[]
After
previously seen=[6];new vertices=[]
Conditions: none
At each layer inspect every neighbor of every frontier vertex,then add only unvisited distinct labels.Do not stop at a plausible-looking fragment.
Step identifiers and dependencies
Rule: component_complete_frontier_expansion · Uses: G001, S002
S004 · implies
Before
selected vertex:final visited=[6];all [vertex,neighbors,outside neighbors] rows=[[6, [], []]]
After
no edge leaves visited;every visited vertex has a path;visited equals full reachable set
Conditions: none
Path witnesses give visited⊆reachable.At completion no edge leaves visited.If a path reached outside,it would have a first leaving edge,a contradiction,so reachable⊆visited.Both directions establish the whole component,not a traversal order.
Step identifiers and dependencies
Rule: component_reachable_and_closed_set_completeness · Uses: G001, S003