Abstract

We consider 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. (Split graphs are obtained by setting k=ℓ=1.) 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 main result is a forbidden subgraph characterization of chordal (k,ℓ)-graphs. We also give an O(n(m+n)) recognition algorithm for chordal (k,ℓ)-graphs. When k=1, our algorithm runs in time O(m+n).

In particular, we obtain a new simple and efficient greedy algorithm for the recognition of split graphs, from which it is easy to derive the well known forbidden subgraph characterization of split graphs. The algorithm and the characterization extend, in a natural way, to the ‘list’ (or ‘pre-colouring extension’) version of the split partition problem—given a graph with some vertices pre-assigned to the independent set, or to the clique, is there a split partition extending this pre-assignment? Another way to think of our main result is the following min–max property of chordal graphs: for each integer r1, the maximum number of independent K_r's (i.e., of vertex disjoint subgraphs of G, each isomorphic to K_r, with no edges joining two of the subgraphs) equals the minimum number of cliques of G that meet all the K_r's of G.

Author Keywords: Chordal graphs; Split graphs; Min–max theorems; Greedy algorithms; Pre-colouring extension; List partitions

Article Outline

1. Introduction

2. The theorems

3. The algorithms

4. The case of one independent set, emphasizing split graphs

5. Pre-colouring extension

References