On counting interval lengths of interval graphs

Given an interval graph GG, the interval count problem is that of computing the minimum number IC(G)IC(G) of interval lengths needed to represent GG. Although the problem of deciding whether IC(G)=1IC(G)=1 is equivalent to that of recognizing unit-interval graphs, which is a well-known problem having several efficient recognition approaches, very little is known about deciding efficiently whether IC(G)=kIC(G)=k for fixed k≥2k≥2. We provide efficient computations of the interval count of generalizations of threshold graphs.