Homomorphism complexes, reconfiguration, and homotopy for directed graphs

Abstract

The neighborhood complex of a graph was introduced by Lovász to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such ‘Hom complexes’ are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs (digraphs). For any pair of digraphs graphs G and H, we consider the polyhedral complex Hom⃗(G,H) that parametrizes the directed graph homomorphisms f:G→H. This construction can be seen as a special case of the poset structure on the set of multihomomorphisms in more general categories, as introduced by Kozlov, Matsushita, and others. Hom complexes of digraphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals.We study examples of directed Hom⃗ complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a digraph and prove that its homotopy type is recovered as the Hom⃗ complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a digraph version of the Mycielski construction, as well as vanishing theorems for higher homology. The Hom⃗ complexes of digraphs provide a natural framework for reconfiguration of homomorphisms of digraphs. Inspired by notions of directed graph colorings we study the connectivity of Hom⃗(G,Tn) for Tn a tournament, obtaining a complete answer for the case of transitive Tn. If G=Tm is also a transitive tournament, we describe a connection to mixed subdivisions of dilated simplices. Finally we use paths in the internal hom objects of digraphs to define various notions of homotopy, and discuss connections to the topology of Hom⃗ complexes.