Graph Homomorphism, Monotone Classes and Bounded Pathwidth

Abstract

In recent work by Johnson et al. (2022), a framework was described for the study of graph problems over classes specified by omitting each of a finite set of graphs as subgraphs. If a problem falls into the framework then its computational complexity can be described for all such graph classes, giving a dichotomy between those classes for which the problem is hard and those for which it is easy. In this article, we consider several variants of the homomorphism problem in relation to this framework. It is known that certain homomorphism problems, e.g. $C_5$-Colouring, do not sit in the framework. By contrast, we show that the more general problem of Graph Homomorphism does sit in the framework, with hard cases NP-complete and easy cases in P. We go on to consider several locally constrained variants of the homomorphism problem, namely the locally bijective, surjective and injective variants. Like $C_5$-Colouring, none of these is in the framework. However, where a bounded-degree restrictions are considered, we prove that each of these problems is in our framework, with hard cases NP-complete and easy cases in P Next, we give the first example of a problem in the framework such that hardness is in the polynomial hierarchy above NP. This comes from a list colouring game, realised through first-order logic as quantified constraints. We show that with the additional restriction of bounded alternation, the problem is contained in the framework. The hard cases are $\Pi_{2k}^\mathrm{P}$-complete and the easy cases are in P. Finally, we go on to consider an aforementioned problem from our framework, complete for the second level of the polynomial hierarchy, under the omission in the input of not just a graph, but rather a graph $H$ annotated with the types for each vertex: existential or universal.