Arbitrarily Partitionable {2K(2), C-4}-Free Graphs

A graph G = (V, E) of order n is said to be arbitrarily partitionable if for each sequence lambda = (lambda(1), lambda(2), horizontal ellipsis , lambda(p)) of positive integers with lambda(1) +center dot horizontal ellipsis center dot+lambda(p) = n, there exists a partition (V-1, V-2, horizontal ellipsis , V-p) of the vertex set V such that V-i induces a connected subgraph of order lambda(i) in G for each i is an element of {1, 2, horizontal ellipsis , p}. In this paper, we show that a threshold graph is arbitrarily partitionable if and only if it admits a perfect matching or a near perfect matching. We also give a necessary and sufficient condition for a {2K(2), C-4}-free graph being arbitrarily partitionable, as an extension for a result of Broersma, Kratsch and Woeginger [Fully decomposable split graphs, European J. Combin. 34 (2013) 567-575] on split graphs.