  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 $\beta(G)$ of $G$ is the largest integer $k$ such that $G$ has a
$k$-block.
  We investigate how $\beta$ 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 $\beta(G)$ has a dual width invariant, the
block-width ${\rm bw}(G)$ of $G$. Our algorithms imply the duality theorem
$\beta = {\rm bw}$: a graph has a block-decomposition of width and adhesion $<
k$ if and only if it contains no $k$-block.

Combinatorics