Toughness of Recursively Partitionable Graphs

Abstract

A simple graph G = (V,E) on n vertices is said to be recursively partitionable (RP) if G ≃ K1, or if G is connected and satisfies the following recursive property: for every integer partition a1, a2, . . . , ak of n, there is a partition {A1,A2, . . . ,Ak} of V such that each |Ai| = ai, and each induced subgraph G[Ai] is RP (1 ≤ i ≤ k). We show that if S is a vertex cut of an RP graph G with |S| ≥ 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.

Keywords

• recursively partitionable graph
• rp graph
• recursively vertex decomposable graph
• ravd graph
• toughness