Testing list H-homomorphisms

Abstract

In the List H- Homomorphism Problem, for a graph H that is a parameter of the problem, an instance consists of an undirected graph G with a list constraint ${L(v) \subseteq V(H)}$ for each variable ${v \in V(G)}$, and the objective is to determine whether there is a list H-homomorphism ${f:V(G) \to V(H)}$, that is, ${f(v) \in L(v)}$ for every ${v \in V(G)}$ and ${(f(u),f(v)) \in E(H)}$ whenever ${(u,v) \in E(G)}$.

We consider the problem of testing list H-homomorphisms in the following weighted setting: An instance consists of an undirected graph G, list constraints L, weights imposed on the vertices of G, and a map ${f:V(G) \to V(H)}$ given as an oracle access. The objective is to determine whether f is a list H-homomorphism or far from any list H-homomorphism. The farness is measured by the total weight of vertices ${v \in V(G)}$ for which f(v) must be changed so as to make f a list H-homomorphism. In this paper, we classify graphs H with respect to the number of queries to f required to test the list H-homomorphisms. Specifically, we show that (i) list H-homomorphisms are testable with a constant number of queries if and only if H is a reflexive complete graph or an irreflexive complete bipartite graph and (ii) list H-homomorphisms are testable with a sublinear number of queries if and only if H is a bi-arc graph.