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.

Let p and q be positive integers with p/q ≥ 2. The "reconfiguration problem" for circular colourings asks, given two (p, q)-colourings f and g of a graph G, is it possible to transform f into g by changing the colour of one vertex at a time such that every intermediate mapping is a (p, q)-colouring? We show that this problem can be solved in polynomial time for 2 ≤ p/q < 4 and that it is PSPACE-complete for p/q ≥ 4. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings. As an application of the reconfiguration algorithm, we show that graphs with fewer than (k − 1)!/2 cycles of length divisible by k are k-colourable.

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The complexity of the list homomorphism problem for signed graphs appears difficult to classify. Existing results focus on special classes of signed graphs, such as trees [4] and reflexive signed graphs [25]. Irreflexive signed graphs are in a certain sense the heart of the problem, as noted by a recent paper of Kim and Siggers. We focus on a special class of irreflexive signed graphs, namely those in which the unicoloured edges form a spanning path or cycle, which we call separable signed graphs. We classify the complexity of list homomorphisms to these separable signed graphs; we believe that these signed graphs will play an important role for the general resolution of the irreflexive case. We also relate our results to a conjecture of Kim and Siggers concerning the special case of weakly balanced irreflexive signed graphs; we have proved the conjecture in another paper, and the present results add structural information to that topic.

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This work brings together ideas of mixing graph colorings, discrete homotopy, and precoloring extension. A particular focus is circular colorings. We prove that all the (k,q)‐colorings of a graph G can be obtained by successively recoloring a single vertex provided k/q≥2col(G) along the lines of Cereceda, van den Heuvel, and Johnson's result for k‐colorings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colorings coincide. As a corollary, we obtain an Albertson‐type extension theorem for (k,q)‐precolorings of circular cliques. Such a result was first conjectured by Albertson and West. General results on homomorphism mixing are presented, including a characterization of graphs G for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role.

A dominating broadcast on a graph G = (V E) is a function : V → {0 1 diam G} such that ( ) ≤ ( ) (the eccentricity of ) for all ∈ V and such that each vertex is within distance ( ) from a vertex with ( ) > 0. The cost of a broadcast is σ ( ) = ∈V ( ), and the broadcast number γ b (G) is the minimum cost of a dominating broadcast. A set X ⊆ V (G) is said to be irredundant if each ∈ X dominates a vertex that is not dominated by any other vertex in X ; possibly = . The irredundance number ir(G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound γ b (G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir(G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for γ b . MSC:05C69, 05C70

Abstract. Let H be a graph and k ≥ 3. A near-unanimity function of arity k is a mapping g from the k-tuples over V (H) to V (H) such that g(x 1 , x 2 , . . . , x k ) is adjacent to g(x 1 , x 2 , . . . , x k ) whenever x i x i ∈ E(H) for each i = 1, 2, . . . , k, and g(x 1 , x 2 , . . . , x k ) = a whenever at least k − 1 of the x i 's equal a. Feder and Vardi proved that, if a graph H admits a near-unanimity function, then the homomorphism extension (or retraction) problem for H is polynomial time solvable. We focus on near-unanimity functions on reflexive graphs. The best understood are reflexive chordal graphs H: they always admit a near-unanimity function. We bound the arity of these functions in several ways related to the size of the largest clique and the leafage of H, and we show that these bounds are tight. In particular, it will follow that the arity is bounded by n − √ n + 1, where n = |V (H)|. We investigate substructures forbidden for reflexive graphs that admit a near-unanimity function. It will follow, for instance, that no reflexive cycle of length at least four admits a near-unanimity function of any arity. However, we exhibit nonchordal graphs which do admit near-unanimity functions. Finally, we characterize graphs which admit a conservative near-unanimity function. This characterization has been predicted by the results of Feder, Hell, and Huang. Specifically, those results imply that, if P = NP, the graphs with conservative near-unanimity functions are precisely the so-called bi-arc graphs. We give a proof of this statement without assuming P = NP.