Interval degree and bandwidth of a graph

The interval degree id(G) of a graph G is defined to be the smallest max-degree of any interval supergraphs of G. One of the reasons for our interest in this parameter is that the bandwidth of a graph is always between id(G)/2 and id(G). We prove also that for any graph G the interval degree of G is at least the pathwidth of G2. We show that if G is an AT-free claw-free graph, then the interval degree of G is equal to the clique number of G2 minus one. Finally, we show that there is a polynomial time algorithm for computing the interval degree of AT-free claw-free graphs.

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