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A note of arbitrarily vertex decomposable graphs

Marczyk A.

Opuscula Mathematica

2006

Vol. 26, no. 1

109-118

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n1,..., nk) of positive integers such that n1 + ... + nk = n there exists a partition (V1,..., Vk) of the vertex set of G such that for each i ∈ {1,..., k}, Vi induces a connected subgraph of G on ni vertices. In this paper we show that if G is a two-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. We present another result for connected graphs satisfying a similar condition, where the bound n - 3 is replaced by n - 2.