H-colouring Problem Research Articles

Surjective H-colouring: New hardness results

A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f(u) and f(v) of H whenever there is an edge between vertices u and v of G. The H-Colouring problem is to decide if a graph G allows a homomorphism to a fixed graph H. We continue a study on a variant of this problem, namely the Surjective H-Colouring problem, which imposes the homomorphism to be vertex-surjective. We build upon previous results and show that this problem is NP-complete for every connected graph H that has exactly two vertices with a self-loop as long as these two vertices are not adjacent. As a result, we can classify the computational complexity of Surjective H-Colouring for every graph H on at most four vertices.

Weak near-unanimity functions and digraph homomorphism problems

For a fixed digraph H, the problem of deciding whether there exists a homomorphism from an input digraph G to H is known as the H-colouring problem. An algebraic approach to this problem was pioneered by Jeavons et al. in the context of the more general constraint satisfaction problem. Results by Larose and Zádori and by Maróti and McKenzie allow one to interpret the algebraic approach in terms of so called weak near-unanimity functions (WNUFs). In this paper, we focus on weak near-unanimity functions and how they apply to the H-colouring problem in particular. Our results range from non-existence results of WNUFs for certain digraphs H, to the existence of WNUFs for the well known polynomial cases of the H-colouring problem treated by Gutjahr, Woeginger and Welzl. Along the way we develop WNUF analogs of the indicator and sub-indicator constructions of Hell and Nešetřil. These results provide evidence to the conjecture that weak near-unanimity functions are the right measure for the complexity of the H-colouring problem.

Two new homomorphism dualities and lattice operations

The study of constraint satisfaction problems (CSPs) definable in various fragments of Datalog has recently gained considerable importance.We consider CSPs that are definable in the smallest natural recursive fragment of Datalog—monadic linear Datalog with at most one EDB (extensional database predicates) per rule, and also in the smallest nonlinear extension of this fragment. We give combinatorial and algebraic characterizations of such problems, in terms of homomorphism dualities and lattice operations, respectively.We then apply our results to study graph H-colouring problems.

The complexity of partition functions

We give a complexity theoretic classification of the counting versions of so-called H-colouring problems for graphs H that may have multiple edges between the same pair of vertices. More generally, we study the problem of computing a weighted sum of homomorphisms to a weighted graph H. The problem has two interesting alternative formulations: first, it is equivalent to computing the partition function of a spin system as studied in statistical physics. And second, it is equivalent to counting the solutions to a constraint satisfaction problem whose constraint language consists of two equivalence relations. In a nutshell, our result says that the problem is in polynomial time if the adjacency matrix of H has row rank 1, and #P-hard otherwise.

Greedy algorithms, H-colourings and a complexity-theoretic dichotomy

Let H be a fixed undirected graph. An H-colouring of an undirected graph G is a homomorphism from G to H. If the vertices of G are partially ordered then there is a generic non-deterministic greedy algorithm which computes all lexicographically first maximal H-colourable subgraphs of G. We show that the complexity of deciding whether a given vertex of G is in a lexicographically first maximal H-colourable subgraph of G is NP-complete, if H is bipartite, and Σ2p-complete, if H is non-bipartite. This result complements Hell and Nešetřil's seminal dichotomy result that the standard H-colouring problem is in P, if H is bipartite, and NP-complete, if H is non-bipartite. Our proofs use the basic techniques established by Hell and Nešetřil, combinatorially adapted to our scenario.

The complexity of H-colouring of bounded degree graphs

We investigate the complexity of the H-colouring problem restricted to graphs of bounded degree. The H-colouring problem is a generalization of the standard c-colouring problem, whose restriction to bounded degree graphs remains NP-complete, as long as c is smaller than the degree bound (otherwise we can use the theorem of Brooks to obtain a polynomial time algorithm). For H-colouring of bounded degree graphs, while it is also the case that most problems are NP-complete, we point out that, surprisingly, there exist polynomial algorithms for several of these restricted colouring problems. Our main objective is to propose a conjecture about the complexity of certain cases of the problem. The conjecture states that for graphs of chromatic number three, all situations which are not solvable by the colouring algorithm inherent in the theorem of Brooks are NP-complete. We motivate the conjecture by proving several supporting results.

Duality and Polynomial Testing of Tree Homomorphisms

Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. We prove that if H has tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP -complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P 6= NP , then no oriented triad H with an NP -complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP -complete Hcolouring problems given in the companion paper has tree duality.

Hereditarily hard H-colouring problems

Let H be a graph (respectively digraph) whose vertices are called ‘colours’. An H-colouring of a graph (respectively digraph) G is an assignment of these colours to the vertices of G so that if u is adjacent to v in G, then the colour of u is adjacent to the colour of v in H. We continue the study of the complexity of the H-colouring problem ‘Does a given graph (respectively digraph) admit an H-colouring?’. For graphs it was proved that the H-colouring problem is NP-complete whenever H contains an odd cycle, and is polynomial for bipartite graphs. For directed graphs the situation is quite different, as the addition of an edge to H can result in the complexity of the H-colouring problem shifting from NP-complete to polynomial. In fact, there is not even a plausible conjecture as to what makes directed H-colouring problems difficult in general. Some order may perhaps be found for those digraphs H in which each vertex has positive in-degree and positive out-degree. In any event, there is at least, in this case, a conjecture of a classification by complexity of these directed H-colouring problems. Another way, which we propose here, to bring some order to the situation is to restrict our attention to those digraphs H which, like odd cycles in the case of graphs, are hereditarily hard, i.e., are such that the H′-colouring problem is NP-hard for any digraph H′ containing H as a subdigraph. After establishing some properties of the digraphs in this class, we make a conjecture as to precisely which digraphs are hereditarily hard. Surprisingly, this conjecture turns out to be equivalent to the one mentioned earlier. We describe several infinite families of hereditarily hard digraphs, and identify a family of digraphs which are minimal in the sense that it would be sufficient to verify the conjecture for members of that family.

The complexity of colouring symmetric relational systems

A relational system, S, is a set T with relations R 1, R 2, …, R k on T, denoted S = ( T, R 1, R 2, …, R k ). We consider relational systems where all the relations are binary, symmetric and antireflexive. The underlying graph, G, of a relational systems S = ( T, R 1, …, R k ) is the graph with vertex set V( G) = T and uv ∈( G) if ( u, v) ∈ R i for some i. We use the terms path (respectively) tree) to mean a relational system whose underlying graph is a path (respectively tree). Let E i ( G) = { uv|( u, v) ∈ R i } be the set of edges coloured i. The relational system defined by ( T, R 1, R 2, …, R k ) can be represented in the form of a coloured graph V( G) together with E 1( G), E 2( G), …, E k ( G). A homomorphism G → H is a mapping to V( G) to V( H) that preserves edge colours. Such a mapping is called an H-colouring of G. For a fixed H, the H-colouring problem is: H - COL ( H-colouring). Instance: A relational system G. Question: Does there exist an H-colouring of G? We show that H-COL is polynomial when H is a path and we show that trees exist such that H-COL is NP-complete. These results parallel the results of directed graphs concerning oriented paths and trees. However, the trees presented here are somewhat smaller and simpler than the known examples for directed graphs.

On the complexity of colouring by superdigraphs of bipartite graphs

Let H be a directed graph whose vertices are called colours. An H-colouring of a digraph G is an assignment of these colours to the vertices of G so that if g is adjacent to g′ in G then colour(g) is adjacent to colour(g′) in H (i.e., a homomorphism G→H). In this paper we continue the study of the H-colouring problem, that is, the decision problem ‘Is there an H-colouring of a given digraph G?’ It follows from a result of Hell and Nešetřil that this problem is NP-complete whenever H contains a symmetric odd cycle. We consider digraphs for which the symmetric part of H is bipartite, that is, digraphs H which can be constructed from the equivalence digraph of an undirected bipartite graph by adding some arcs. We establish some sufficient conditions for these H-colouring problems to be NP-complete. A complete classification is established for the subclass of ‘partitionable digraphs’, which we introduce.

On the Complexity of Colouring by Vertex-Transitive and Arc-Transitive Digraphs

Let H be a fixed directed graph whose vertices are called colours. An H-colouring of a digraph G is an assignment of these colours to the vertices of G such that if x is adjacent to y in G, then colour$( x )$ is adjacent to colour$( y )$ in H (i.e., a homomorphism$G \to H$). In this paper the complexity of the H-colouring problem, when the directed graph H is vertex-transitive or arc-transitive, is investigated. In both instances a complete classification is obtained.

The effect of two cycles on the complexity of colourings by directed graphs

Let H be a fixed directed graph. An H-colouring of a directed graph D is a mapping f: V( D)→ V( H) such that f( x) f( y) is an edge of H whenever xy is an edge of D. We study the following H-colouring problem: Instance: A directed graph D. Question: Does there exist an H-colouring of D? In an earlier paper [2] it is shown that among semicomplete digraphs H, it is the existence of two directed cycles in H which makes the H-colouring problem (NP-) hard. In this paper we provide further classes of digraphs in which two directed cycles in H make the H-colouring problem NP-hard. These include both classes of dense and of sparse digraphs. There still appears to be no natural conjecture as to what digraphs H give NP-hard H-colouring problems; however, in view of our results, we are led to make such a conjecture for digraphs without sources and sinks.

The Complexity of Colouring by Semicomplete Digraphs

The following problem, known as the H-colouring problem, is studied. An H-colouring of a directed graph D is a mapping $f:V( D ) \to V( H )$ such that $( f( x ),f( y ) )$ is an edge of H whenever $( x,y )$ is an edge of D. The H-colouring problem is the following. Instance: A directed graph D. Question: Does there exist an H-colouring of D? In this paper it is shown that for semicomplete digraphs T the T-colouring problem is NP-complete when T has more than one directed cycle, and polynomially decidable otherwise.