$k$-Blocks: a connectivity invariant for graphs

Abstract

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by fewer than $k$ other vertices. The block number $β(G)$ of $G$ is the largest integer $k$ such that $G$ has a $k$-block. We investigate how $β$ interacts with density invariants of graphs, such as their minimum or average degree. We further present algorithms that decide whether a graph has a $k$-block, or which find all its $k$-blocks. The connectivity invariant $β(G)$ has a dual width invariant, the block-width ${\rm bw}(G)$ of $G$. Our algorithms imply the duality theorem $β= {\rm bw}$: a graph has a block-decomposition of width and adhesion $< k$ if and only if it contains no $k$-block.

22 pages, 5 figures. This is an extended version the journal article, which has by now appeared. The version here contains an improved version of Theorem 5.3 (which is now best possible) and an additional section with examples at the end