Random graphs from a block-stable class

Abstract

A class of graphs is called block-stable when a graph is in the class if and only if each of its blocks is. We show that, as for trees, for most $n$-vertex graphs in such a class, each vertex is in at most $(1+o(1))\log n/\log \log n$ blocks, and each path passes through at most $5(n\log n{)}^{1/2}$ blocks. These results extend to `weakly block-stable' classes of graphs.