Interval degree and bandwidth of a graph

The interval degree id(G) of a graph G is de/ned 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 G. 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 G minus one. Finally, we show that there is a polynomial time algorithm for computing the interval degree of AT-free claw-free graphs. ? 2003 Elsevier B.V. All rights reserved. MSC: 05C78; 05C85; 68R10