Correspondence Homomorphisms to Reflexive Graphs

Journal of Combinatorial Theory, Series B, 1998

Let H be a fixed graph. We introduce the following list homomorphism problem: Given an input graph G and for each vertex

Discrete Applied Mathematics, 2006

Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected input graph D admits a homomorphism to H . The list homomorphism problem for H is a generalization of the homomorphism problem for H, where every vertex x ∈ V (D) is assigned a set L x of possible colors (vertices of H).

Lecture Notes in Computer Science, 2008

For digraphs G and H, a homomorphism of G to H is a mapping f : V (G)→V (H) such that uv ∈ A(G) implies f (u)f (v) ∈ A(H). If moreover each vertex u ∈ V (G) is associated with costs c i (u), i ∈ V (H), then the cost of a homomorphism f is u∈V (G) c f (u) (u). For each fixed digraph H, the minimum cost homomorphism problem for H, denoted MinHOM(H), is the following problem. Given an input digraph G, together with costs c i (u), u ∈ V (G), i ∈ V (H), and an integer k, decide if G admits a homomorphism to H of cost not exceeding k. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as chromatic partition optimization and applied problems in repair analysis. For undirected graphs the complexity of the problem, as a function of the parameter H, is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. We focus on the minimum cost homomorphism problem for reflexive digraphs H (every vertex of H has a loop). It is known that the problem MinHOM(H) is polynomial time solvable if the digraph H has a Min-Max ordering, i.e., if its vertices can be linearly ordered by < so that i < j, s < r and ir, js ∈ A(H) imply that is ∈ A(H) and jr ∈ A(H). We give a forbidden induced subgraph characterization of reflexive digraphs with a Min-Max ordering; our characterization implies a polynomial time test for the existence of a Min-Max ordering. Using this characterization, we show that for a reflexive digraph H which does not admit a Min-Max ordering, the minimum cost homomorphism problem is NP-complete, as conjectured by Gutin and Kim. Thus we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for reflexive digraphs.

Taiwanese Journal of Mathematics, 1999

in Febuary 1999. We survey results related to structural aspects of graph homomorphism. Our aim is to demonstrate that this forms today a compact collection of results and methods which perhaps deserve its name : structural combinatorics. Due to space limitations we concentrate on a sample of areas only: representation of algebraic structures by combinatorial ones (graphs), the poset of colour classes and corresponding algorithmic questions which lead to homomorphism dualities, blending algebraic and complexity approaches.

SIAM Journal on Discrete Mathematics, 2012

The minimum cost homomorphism problem has arisen as a natural and useful optimization problem in the study of graph (and digraph) coloring and homomorphisms: it unifies a number of other well studied optimization problems. It was shown by Gutin, Rafiey, and Yeo that the minimum cost problem for homomorphisms to a digraph H that admits a so-called extended Min-Max ordering is polynomial time solvable, and these authors conjectured that for all other digraphs H the problem is NP-complete. In a companion paper, we gave a forbidden structure characterization of digraphs that admit extended Min-Max orderings. In this paper, we apply this characterization to prove Gutin's conjecture.

European Journal of Combinatorics, 2008

For graphs G and H, a mapping f : V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f (u)f (v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs c i (u), i ∈ V (H), then the cost of the homomorphism f is u∈V (G) c f (u) (u). For each fixed graph H, we have the minimum cost homomorphism problem, written as MinHOM(H). The problem is to decide, for an input graph G with costs c i (u), u ∈ V (G), i ∈ V (H), whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We prove a dichotomy of the minimum cost homomorphism problems for graphs H, with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM(H) is polynomial time solvable. In all other cases the problem MinHOM(H) is NP-hard. This solves an open problem from an earlier paper.

Discrete Applied Mathematics, 2010

For digraphs D and H, a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f (u)f (v) ∈ A(H). For a fixed digraph H, the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM(H). An optimization version of the homomorphism problem was motivated by a realworld problem in defence logistics and was introduced in [13]. If each vertex u ∈ V (D) is associated with costs c i (u), i ∈ V (H), then the cost of the homomorphism f is u∈V (D) c f (u) (u). For each fixed digraph H, we have the minimum cost homomorphism problem for H and denote it as MinHOM(H). The problem is to decide, for an input graph D with costs c i (u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM(H) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM(H) were proved when H is a semicomplete digraph [10], and a semicomplete multipartite digraph [12, 11]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in [9].

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we prove the algebraic CSP dichotomy conjecture holds for digraphs whose symmetric closure is a complete graph, and observe a collapse in the applicability of algorithms for CSPs over directed graphs with both a total source and a total sink: the corresponding CSP is solvable by the "few subpowers algorithm" if and only if it is solvable by a local consistency check algorithm. Moreover, we find that the property of "strict width" and solvability by few subpowers are unstable under first order reductions.

Discrete Mathematics, 2017

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.

arXiv (Cornell University), 2007

For digraphs D and H, a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f (u)f (v) ∈ A(H). For a fixed digraph H, the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM(H). An optimization version of the homomorphism problem was motivated by a realworld problem in defence logistics and was introduced in [12]. If each vertex u ∈ V (D) is associated with costs c i (u), i ∈ V (H), then the cost of the homomorphism f is u∈V (D) c f (u) (u). For each fixed digraph H, we have the minimum cost homomorphism problem for H and denote it as MinHOM(H). The problem is to decide, for an input graph D with costs c i (u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM(H) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM(H) were proved when H is a semicomplete digraph [9], and a semicomplete multipartite digraph [11, 10]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in [8]. 1