An Ore-type condition for arbitrarily vertex decomposable graphs

Abstract

Let G be a graph of order n and r , 1 ≤ r ≤ n , a fixed integer. G is said to be r -vertex decomposable if for each sequence ( n 1 , … , n r ) of positive integers such that n 1 + ⋯ + n r = n there exists a partition ( V 1 , … , V r ) of the vertex set of G such that for each i ∈ { 1 , … , r } , V i induces a connected subgraph of G on n i vertices. G is called arbitrarily vertex decomposable if it is r -vertex decomposable for each r ∈ { 1 , … , n } . In this paper we show that if G is a connected graph on n vertices with the independence number at most ⌈ n / 2 ⌉ and such that the degree sum of any pair of non-adjacent vertices is at least n − 3 , then G is arbitrarily vertex decomposable or isomorphic to one of two exceptional graphs. We also exhibit the integers r for which the graphs verifying the above degree-sum condition are not r -vertex decomposable.