A signed graph [G, ] is a graph G together with an assignment of signs + and − to all the edges of G where is the set of negative edges. Furthermore [G, 1 ] and [G, 2 ] are considered to be equivalent if the symmetric difference of 1 and 2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere-zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph homomorphisms where he showed how

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.

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to find a subset of vertices, normally called a solution set, using which all vertices of the graph are distinguished. The identification can be done by considering the neighborhood within the solution set, or by employing the distances to the solution vertices. Normally the goal is to minimize the size of the solution set then. Here we study the case of interval graphs, unit interval graphs, (bipartite) permutation graphs and cographs. For these classes of graphs we give tight lower bounds for the size of such solution sets depending on the order of the input graph. While such lower bounds for the general class of graphs are in logarithmic order, the improved bounds in these special classes are of the order of either quadratic root or linear in terms of number of vertices. Moreover, the results for cographs lead to linear-time algorithms to solve the considered problems on inputs that are cographs. * u, v if it (totally) dominates exactly one of them. A set S (totally) separates the vertices of a set X if all pairs of X are (totally) separated by a vertex of S. Whenever it is clear from the context, we will only say "separate" and omit the word "totally". We have the three key definitions, that merge the concepts of (total) domination and (total) separation:Definition 1 (Slater [33,34]). A set S of vertices of a graph G is a locating-dominating set if it is a dominating set and it separates the vertices of V (G) \ S. The smallest size of a locating-dominating set of G is the location-domination number of G, denoted γ LD (G). Without the domination constraint, this concept has also been used under the name distinguishing set in [2] and sieve in [28].Definition 2 (Karpovsky, Chakrabarty and Levitin [27]). A set S of vertices of a graph G is an identifying code if it is a dominating set and it separates all vertices of V (G). The smallest size of an identifying code of G is the identifying code number of G, denoted γ ID (G). Definition 3 (Seo and Slater [31]). A set S of vertices of a graph G is an open locating-dominating set if it is a total dominating set and it totally separates all vertices of V (G). The smallest size of an open locating-dominating set of G is the open location-domination number of G, denoted γ OLD (G). This concept has also been called identifying open code in [25]. Separation could also be done using distances from the members of the solution set. Let d(x, u) denote the distance between two vertices x and u. Definition 4 (Harary and Melter [24], Slater [32]). A set B of vertices of a graph G is a resolving set if for each pair u, v of distinct vertices, there is a vertex x of B with d(x, u) = d(x, v). 1 The smallest size of a resolving set of G is the metric dimension of G, denoted dim(G).