The Complexity of Homomorphisms of Signed Graphs and Signed Constraint Satisfaction

A signed graph (

G

, Σ) is an undirected graph

G

together with an assignment of signs (positive or negative) to all its edges, where Σ denotes the set of negative edges. Two signatures are said to be equivalent if one can be obtained from the other by a sequence of resignings (i.e. switching the sign of all edges incident to a given vertex). Extending the notion of usual graph homomorphisms, homomorphisms of signed graphs were introduced, and have lead to some extensions and strengthenings in the theory of graph colorings and homomorphisms. We study the complexity of deciding whether a given signed graph admits a homomorphism to a fixed target signed graph [

H

,Σ], i.e. the (

H

,Σ)

-Coloring

problem. We prove a dichotomy result for the class of all (

C

k

,Σ)

-Coloring

problems (where

C

k

is a cycle of length

k

 ≥ 3): (

C

k

,Σ)

-Coloring

is NP-complete, unless both

k

and the size of Σ are even. We conjecture that this dichotomy can be extended to all signed graphs in a natural way. We also introduce the more general concept of signed constraint satisfaction problems and show that a dichotomy for such problems is equivalent to the statement of the Feder-Vardi Dichotomy Conjecture.