List Homomorphisms and Circular Arc Graphs

Abstract

, H, and lists \(\), a list homomorphism of G to Hwith respect to the listsL is a mapping \(\), such that \(\) for all \(\), and \(\) for all \(\). The list homomorphism problem for a fixed graph H asks whether or not an input graph G together with lists \(\), \(\), admits a list homomorphism with respect to L. We have introduced the list homomorphism problem in an earlier paper, and proved there that for reflexive graphs H (that is, for graphs H in which every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP-complete otherwise. Here we consider graphs H without loops, and find that the problem is closely related to circular arc graphs. We show that the list homomorphism problem is polynomial time solvable if the complement of H is a circular arc graph of clique covering number two, and is NP-complete otherwise. For the purposes of the proof we give a new characterization of circular arc graphs of clique covering number two, by the absence of a structure analogous to Gallai's asteroids. Both results point to a surprising similarity between interval graphs and the complements of circular arc graphs of clique covering number two.