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Finite maximal extension of a 2-connected subgraph
Statement
Every 2-connected subgraph of a finite simple graph is contained in a maximal 2-connected subgraph, hence in a nontrivial block.
Facts & Assumptions
A finite simple graph has finite vertex and edge sets, and its subgraphs are specified by subsets of these sets. A finite simple graph is a finite vertex set together with a set of two-element vertex subsets Subgraphs, induced subgraphs and spanning subgraphs
A maximal 2-connected subgraph is a nontrivial block; maximality is with respect to subgraph containment. Cutvertices, 2-connected pieces and nontrivial blocks
Proof
Given: The graphs and hypotheses in the Statement.
Goal: The asserted conclusion.
Let be the given 2-connected subgraph. There are finitely many subgraphs of the ambient graph, so the nonempty family of 2-connected subgraphs containing is finite. Choose one maximizing the sum of its vertex and edge counts.
A proper supergraph in that family would strictly increase that sum. There is therefore no larger 2-connected subgraph containing the chosen one, and it is a nontrivial block containing .
Depends on
Dependency tree · 2 levels
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