On the Complete Width and Edge Clique Cover Problems

Abstract

A complete graph is the graph in which every two vertices are adjacent. For a graph \(G=(V,E)\), the complete width of G is the minimum k such that there exist k independent sets \(\mathtt {N}_i\subseteq V\), \(1\le i\le k\), such that the graph \(G'\) obtained from G by adding some new edges between certain vertices inside the sets \(\mathtt {N}_i\), \(1\le i\le k\), is a complete graph. The complete width problem is to compute the complete width of a given graph. In this paper we study the complete width problem. We show that the complete width problem is NP-hard on \(3K_2\)-free bipartite graphs and polynomially solvable on \(2K_2\)-free bipartite graphs and on \((2K_2,C_4)\)-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on \(K_{2,2,2}\)-free co-bipartite graphs and polynomially solvable on \(K_{2,2}\)-free co-bipartite graphs and on \((2K_2, C_4)\)-free graphs. We also determine all graphs of small complete width \(k\le 3\).