Connectivity for bridge-alterable graph classes

Abstract

A collection of graphs is called bridge-alterable if, for each graph G with a bridge e, G is in the class if and only if G-e is. For example the class of forests is bridge-alterable. For a random forest $F_n$ sampled uniformly from the set of forests on vertex set {1,..,n}, a classical result of Renyi (1959) shows that the probability that $F_n$ is connected is $e^{-1/2 +o(1)}$. Recently Addario-Berry, McDiarmid and Reed (2012) and Kang and Panagiotou (2013) independently proved that, given a bridge-alterable class, for a random graph $R_n$ sampled uniformly from the graphs in the class on {1,..,n}, the probability that $R_n$ is connected is at least $e^{-1/2 +o(1)}$. Here we give a more straightforward proof, and obtain a stronger non-asymptotic form of this result, which compares the probability to that for a random forest. We see that the probability that $R_n$ is connected is at least the minimum over $\frac25 n < t \leq n$ of the probability that $F_t$ is connected.