On the interval number of special graphs

The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals, denoted by i(G). Griggs and West showed that i(G) ≤ [1/2(d + 1)]. We describe the extremal graphs for that inequality when dis even. For three special perfect graph classes we give bounds on the interval number in terms of the independence number. Finally, we show that a graph needs to contain large complete bipartite induced subgraphs in order to have interval number larger than the random graph on the same number of vertices.