$k$-Blocks: a connectivity invariant for graphs

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.

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