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List homomorphisms generalize list colourings in the following way: Given graphs G, H, and lists L(v) The list homomorphism problem for a fixed graph H asks whether or not an input, 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.

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Given graphs G; H, and lists LðvÞ V ðHÞ; v 2 V ðGÞ, a list homomorphism of G to H with respect to the lists L is a mapping f : V ðGÞ ! V ðHÞ such that uv 2 EðGÞ implies f ðuÞf ðvÞ 2 EðHÞ, and f ðvÞ 2 LðvÞ for all v 2 V ðGÞ. The list homomorphism problem for a fixed graph H asks whether or not an input graph G, together with lists ------------------

A digraph is quasi-transitive if there is a complete adjacency between the inset and the outset of each vertex. Quasi-transitive digraphs are interesting because of their relation to comparability graphs. Specifically, a graph can be oriented as a quasi-transitive digraph if and only if it is a comparability graph. Quasi-transitive digraphs are also of interest as they share many nice properties of tournaments. Indeed, w e show that every strongly connected quasi-transitive digraph D on at least four vertices has two vertices u1 and u2 such that D -u, is strongly connected for i = 1,2. A result of tournaments on the existence of a pair of arc-disjoint in-and out-branchings rooted at the same vertex can also be extended to quasi-transitive digraphs. However, some properties of tournaments, like hamiltonicity, cannot be extended directly to quasi-transitive digraphs. Therefore w e characterize those quasi-transitive digraphs which have a hamiltonian cycle, respectively a hamiltonian path. We show the existence of highly connected quasi-transitive digraphs D with a factor (a collection of disjoint cycles covering the vertex set of D), which have a cycle of every length 3 I k I IV(D)l -1 through every vertex and yet they are not hamiltonian. Finally w e characterize pancyclic and vertex pancyclic quasi-transitive digraphs. 0 1995, John Wiley & Sons, Inc.

We prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two, which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hope may prove helpful in finding a more efficient recognition algorithm than presently known. We use these results to show equality, amongst bipartite graphs, of several classes of structured graphs (proper interval bigraphs, complements of proper circular arc graphs, asteroidaltriple-free graphs, permutation graphs, and co-comparability graphs). Our results verify a conjecture of Lundgren and disprove a conjecture of Mü ller.

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