Dense Arbitrarily Vertex Decomposable Graphs

Abstract

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n 1, . . ., n k) of positive integers such that n 1 + · · · + n k = n there exists a partition (V 1, . . ., V k) of the vertex set of G such that for each i ∈{1,...,k}, V i induces a connected subgraph of G on n i vertices. The main result of the paper reads as follows. Suppose that G is a connected graph on n ≥ 20 vertices that admits a perfect matching or a matching omitting exactly one vertex. If the degree sum of any pair of nonadjacent vertices is at least n - 5, then G is arbitrarily vertex decomposable. We also describe 2-connected arbitrarily vertex decomposable graphs that satisfy a similar degree sum condition. © 2011 The Author(s).