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.
['Antoni Marczyk']
An Ore-type condition for arbitrarily vertex decomposable graphs