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. For any pair of directed graphs 𝐺 and 𝐻 we consider the polyhedral complex − −− → Hom(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 directed graphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals.We study examples of directed − −− → Hom 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 directed graph and prove that its homotopy type is recovered as the − −− → Hom 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 directed graph version of the Mycielksi construction, as well as vanishing theorems for higher homology. The − −− → Hom complexes of directed graphs provide a natural framework for reconfiguration of homomorphisms of directed graphs. Inspired by notions of directed graph colorings we study the connectivity of − −− → Hom(𝐺, 𝑇 𝑛 ) for 𝑇 𝑛 a tournament, obtaining a complete answer for the case of transitive 𝑇 𝑛 . If 𝐺 = 𝑇 𝑚 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 directed graphs to define various notions of homotopy, and discuss connections to the topology of − −− → Hom complexes.

Eilenberg-Steenrod homology and cohomology theories for Čech's closure spaces

“…A directed analogue of A-homotopy for simple directed graphs has also been studied [22]. More recently, Dochtermann and Singh, defined several homotopy theories for directed graphs, using both cylinder and path objects [15]. Our work also extends these ideas to closure spaces.…”

Reconfiguration of Digraph Homomorphisms

“…We are aware of two results for H-Recoloring for digraphs H. The first one is by Brewster et al, who showed that H-Recoloring admits a polynomial-time algorithm if H is a reflexive digraph cycle that does not contain a 4-cycle of algebraic girth 0 [3]. In spirit this algorithm uses the topological approach of Wrochna that reduces the task of finding H-recoloring sequences to finding vertex walks in H. Secondly, Dochterman and Singh study the Hom-complex for digraphs G and H and show that it is connected (in the topological sense) if H is a transitive tournament T n [8] on n vertices. From this they conclude that any instance of T n -Recoloring is a YES-instance and give a polynomial-time algorithm that finds a T n -recoloring sequence.…”

“…Despite several positive [1,5,7,16] and negative [2,5,13] results in this setting, a complete classification of the complexity of H-Recoloring for undirected graphs is not known. In the more general context of digraphs, we are aware of only two results for H-Recoloring, which consider the case where H is a transitive tournament [8] and where H is some orientation of a reflexive digraph cycle [3] (reflexive means that there is a loop on each vertex). We extend the topological approach of Wrochna for H-Recoloring for undirected graphs to digraphs and obtain the following results, which generalize the previous algorithmic results in [1,3,7,16].…”