Bounded-Degree Forbidden Patterns Problems Are Constraint Satisfaction Problems

Abstract

Forbidden Patterns problem (FPP) is a proper generalisation of Constraint Satisfaction Problem (CSP). FPP was introduced in [1] as a combinatorial counterpart of MMSNP, a logic which was in turn introduced in relation to CSP by Feder and Vardi [2]. We prove that Forbidden Patterns Problems are Constraint Satisfaction Problems when restricted to graphs of bounded degree. This is a generalisation of a result by Haggkvist and Hell who showed that F-moteness of bounded-degree graphs is a CSP (that is, for a given graph F there exists a graph H so that the class of bounded-degree graphs that do not admit a homomorphism from F is exactly the same as the class of bounded-degree graphs that are homomorphic to H). Forbidden-pattern property is a strict generalisation of F-moteness (in fact of F-moteness combined with a CSP) as it involves both vertex- and edge-colourings of the graph F, and thus allows to express $\mathcal{N}p$-complete problems (while F-moteness is always in $\mathcal{P}$). We finally extend our result to arbitrary relational structures, and prove that every problem in MMSNP, restricted to connected inputs of bounded (hyper-graph) degree, is in fact in CSP.