The complexity of signed graph and edge-coloured graph homomorphisms

Abstract

We study homomorphism problems of signed graphs from a computational point of view. A signed graph (G, Σ) is a graph G where each edge is given a sign, positive or negative; Σ ⊆ E(G) denotes the set of negative edges. Thus, (G, Σ) is a 2-edge-coloured graph with the property that the edge-colours, {+, −}, form a group under multiplication. Central to the study of signed graphs is the operation of switching at a vertex, that results in changing the sign of each incident edge. We study two types of homomorphisms of a signed graph (G, Σ) to a signed graph (H, Π): ec-homomorphisms and s-homomorphisms. Each is a standard graph homomorphism of G to H with some additional constraint. In the former, edge-signs are preserved. In the latter, edge-signs are preserved after the switching operation has been applied to a subset of vertices of G.We prove a dichotomy theorem for s-homomorphism problems for a large class of (fixed) target signed graphs (H, Π). Specifically, as long as (H, Π) does not contain a negative (respectively a positive) loop, the problem is polynomial-time solvable if the core of (H, Π) has at most two edges, and is NP-complete otherwise. (Note that this covers all simple signed graphs.) The same dichotomy holds if (H, Π) has no negative digons, and we conjecture that it holds always. In our proofs, we reduce s-homomorphism problems to certain ec-homomorphism problems, for which we are able to show a dichotomy. In contrast, we prove that a dichotomy theorem for ec-homomorphism problems (even when restricted to bipartite target signed graphs) would settle the dichotomy conjecture of Feder and Vardi.

Complexity of planar signed graph homomorphisms to cycles

“…Extending the Hell-Nešetřil dichotomy [27] for Hom(H) for 1-edge-coloured graph problems, a complexity dichotomy for s-Hom(H) problems was proved in the papers [7,10]. The authors showed that if the s-core of a signed graph H has at least three edges, then s-Hom(H) is NP-complete; it is polynomial-time otherwise.…”

“…An interesting case is the one when H = (C k , σ) is a cycle (k ≥ 3). If H is switching-equivalent either to an all-positive C k , or to an all-negative C k , then the complexity of s-Hom(H) and Planar s-Hom(H) are the same as the ones of Hom(C k ) and Planar Hom(C k ), respectively [7]. When k is odd, one of these two cases holds, and thus by the results of [25,33], in that case Planar s-Hom(H) is NP-complete.…”

“…(More generally, a proper t-colouring of a graph G is a homomorphism to the complete graph K t .) Hom(H) for edge-coloured graphs is studied for example in [5,6,7,9,36].…”

“…We say that two signed graphs G and G ′ are switching-equivalent if G ′ can be obtained from G by any sequence of switchings. Note that one can test switching-equivalence in polynomial time (on labelled graphs), see [7,45].…”

“…In this context, Guenin introduced in [21] a special kind of homomorphisms of signed graphs, whose theory was later developed in [38,39]. Following the terminology in [7], we define an s-homomorphism of a signed graph G to a signed graph H as a vertex-mapping f from V (G) to V (H) such that there exists a signed graph G ′ that is switching-equivalent to G, and f is a classic edge-coloured graph homomorphism of G ′ to H.…”