Edge clique partition in (k, l)-graphs

An edge clique partition of G is a set of edge-disjoint cliques of G which together contain every edge of G. In this work we investigate the computational complexity of determining the minimum size of an edge clique partition of G, denoted by cp(G). This problem is NP-hard for split graphs and K-4-free graphs. We establish a complete complexity classification of this problem on the hierarchy of (k, l)-graphs, graphs that can be partitioned into k independent sets and l complete sets. In addition, we determine in polynomial time the value of cp(G) for interesting subclasses of (k, l)-graphs: split-indifference graphs, partial 2-trees, and triangular grids. (C) 2021 Elsevier B.V. All rights reserved.