Toughness of recursively partitionable graphs

Abstract

A simple graph $G=(V,E)$ on $n$ vertices is said to be recursively partitionable (RP) if $G\simeq K_1$, or if $G$ is connected and satisfies the following recursive property: for every integer partition $a_1,a_2,\dots ,a_k$ of $n$, there is a partition $\{A_1,A_2,\dots ,A_k\}$ of $V$ such that each $|A_i|=a_i$, and each induced subgraph $G[A_i]$ is RP ($1\le i\le k$). We show that if $S$ is a vertex cut of an RP graph $G$ with $|S|\ge 2$, then $G-S$ has at most $3|S|-1$ components. Moreover, this bound is sharp for $|S|=3$. We present two methods for constructing new RP graphs from old. We use these methods to show that for all positive integers $s$, there exist infinitely many RP graphs with an $s$-vertex cut whose removal leaves $2s+1$ components. Additionally, we prove a simple necessary condition for a graph to have an RP spanning tree, and we characterise a class of minimal 2-connected RP graphs.