On generalized split graphs

We prove that in a chordal graph the maximum number of independent (i.e., disjoint and nonadjacent) Krʹs equals the minumum number of cliques that meet all Krʹs. When r = 1, this implies that chordal graphs are perfect. When r = 2, it contains a well known forbidden subgraph characterization of split graphs. We also discuss algorithms for both these problems. In particular, we illustrate the techniques by giving a new simple recognition algorithm for split graphs. We apply these results to the following generalization of split graphs: A graph is said to be a (k, l)-graph if its vertex set can be partitioned into k independent sets and l cliques. Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by (k, l)-graphs in general. (For instance, being a (k, 0)-graph is equivalent to being k-colourable.) However, if we keep the assumption of chordality, nice algorithms and characterization theorems are possible. Indeed, our result gives a forbidden subgraph characterization of (and a polynomial time recognition algorithm for) chordal (k, l)-graphs.