Graph Homomorphism, Monotone Classes and Bounded Pathwidth

TL;DR

The paper investigates a unified framework for classifying the complexity of graph problems on classes defined by omitting a finite set of subgraphs. It shows Graph Homomorphism lies in the $C123$-framework with clear easy/hard dichotomies, while locally constrained homomorphisms are $C23$-problems unless input degree is bounded, in which case they become $C123$-problems. It also extends the framework to quantified constraint problems (QCSPs) with labelled graphs, deriving a nuanced classification that includes $\Pi_{2k}$-alternation results and a sharp boundary between polynomial-time, $\Pi_2^{\mathbf{P}}$-complete, and NP-complete cases. Finally, it analyzes Long Edge Disjoint Paths as a $C23$-problem exhibiting different behavior on bounded treedepth versus bounded pathwidth classes, and outlines open questions on QCSPs over treedepth and labelled omissions. The work broadens understanding of how fixed-structure omissions shape computational complexity across a spectrum of graph problems.

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 $Π_{2k}^\mathrm{P}$-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.

Figures (7)

• Figure 1: Summary of our results.
• Figure 2: Reduction from the proof of Lemma \ref{['lem:graph-hom-weak-C3']}. This is what the $4$-cycle $C_4$ would be reduced to. The first step is to assume each edge is in a triangle. Finally, the edges will be substituted by paths of length $5^r$.
• Figure 3: Self-reduction at the start of the proof of Lemma \ref{['lem:graph-hom-weak-C3-bis']}. This is what the $4$-cycle $C_4$ would be reduced to.
• Figure 4: Construction for clause $C_p$. Dashed lines denote paths of length $2p+1$ with the label corresponding to the list of internal vertices.
• Figure 5: No instance as described in Lemma \ref{['lem:h+p2']}. The subgraph $G_P$ is drawn in bold with existential vertices $e_1,e_2,e_3 \in P$ each with a pair of universal neighbours in $G_P$. The case $a_1=a'_3$ cannot occur as these vertices are distinct in the path $P$.

Theorems & Definitions (49)

• theorem thmcountertheorem: JMOPPSV
• lemma thmcounterlemma: DP89
• lemma thmcounterlemma: GHN00
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• lemma thmcounterlemma: CFHPT15
• lemma thmcounterlemma: CFHPT15