List Homomorphisms and Circular Arc Graphs

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, H, and lists , a list homomorphism of G to H with respect to the lists L 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.

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Received: July 22, 1996/Revised: Revised June 10, 1998

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Feder, T., Hell, P. & Huang, J. List Homomorphisms and Circular Arc Graphs. Combinatorica 19, 487–505 (1999). https://doi.org/10.1007/s004939970003

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  • AMS Subject Classification (1991) Classes:  05C85, 05C75, 05C15; 68R10