On the Equivalence Covering Number of Splitgraphs

An equivalence graph is a disjoint union of cliques. For a graph G let eq(G) be the minimum number of equivalence subgraphs of G needed to cover all edges of G. We call eq( G) the equivalence covering number of G. It was shown in [ 81 that computing the equivalence covering number is NP-hard, even when restricted to graphs in which no two triangles have a vertex in common. We show that the equivalence covering number for splitgraphs can be approximated within an additive constant 1. We also show that obtaining the exact value of the equivalence number of a splitgraph is an NP-hard problem. Using a similar method we also show that it is NP-complete to decide whether the equivalence coveting number of a graph is 3, even for graphs with maximum degree 6 and with maximum clique number 4.