Random graphs from a block-stable class

TL;DR

This paper investigates properties of random graphs within block-stable classes, revealing that most vertices are contained in a logarithmic number of blocks and paths traverse a limited number of blocks, extending to weakly block-stable classes.

Contribution

It establishes bounds on the number of blocks per vertex and per path in random graphs from block-stable classes, generalizing previous results for trees.

Findings

Most vertices are in at most (1+o(1)) log n / log log n blocks.

Paths pass through at most 5 (n log n)^{1/2} blocks.

Results extend to weakly block-stable classes.

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.