On the Complete Width and Edge Clique Cover Problems

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 Ni ⊆ V , 1 ≤ i ≤ k, such that the graph G obtained from G by adding some new edges between certain vertices inside the sets Ni, 1 ≤ i ≤ k, is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most k or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on 3K2-free bipartite graphs and polynomially solvable on 2K2-free bipartite graphs and on (2K2, C4)-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on 3K2-free co-bipartite graphs and polynomially solvable on C4-free co-bipartite graphs and on (2K2, C4)-free graphs. We also give a characterization for k-probe complete graphs which implies that the complete width problem admits a kernel of at most 2 vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most 2 vertices. Finally we determine all graphs of small complete width k ≤ 3.