Computing List Homomorphisms in Geometric Intersection Graphs

Abstract

A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). Let H be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom(H), the instance is a graph G, whose every vertex is equipped with a subset of V(H), called list. We ask if there exists a homomorphism from G to H, such that every vertex from G is mapped to a vertex from its list. We study the complexity of the LHom(H) problem in intersection graphs of various geometric objects. In particular, we are interested in answering the question for what graphs H and for what types of geometric objects, the LHom(H) problem can be solved in time subexponential in the number of vertices of the instance. We fully resolve this question for string graphs, i.e., intersection graphs of continuous curves in the plane. Quite surprisingly, it turns out that the dichotomy coincides with the analogous dichotomy for graphs excluding a fixed path as an induced subgraph [Okrasa, Rzążewski, STACS 2021]. Then we turn our attention to intersections of fat objects. We observe that the (non) existence of subexponential-time algorithms in such classes is closely related to the size $$\textrm{mrc}(H)$$ of a maximum reflexive clique in H, i.e., maximum number of pairwise adjacent vertices, each of which has a loop. We study the maximum value of $$\textrm{mrc}(H)$$ that guarantees the existence of a subexponential-time algorithm for LHom(H) in intersection graphs of (i) convex fat objects, (ii) fat similarly-sized objects, and (iii) disks. In the first two cases we obtain optimal results, by giving matching algorithms and lower bounds.