Graph homomorphisms with infinite targets

AbstractLet H be a fixed graph whose vertices are called colours. Informally, an H-colouring of a graph G is an assignment of these colours to the vertices of G such that adjacent vertices receive adjacent colours. We introduce a new tool for proving NP-completeness of H-colouring problems, which unifies all methods used previously. As an application we extend, to infinite graphs of bounded degree, the theorem of Hell and Nešetřil that classifies finite H-colouring problems by complexity

Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem

The Algebraic Dichotomy Conjecture states that the Constraint Satisfaction Problem over a fixed template is solvable in polynomial time if the algebra of polymorphisms associated to the template lies in a Taylor variety, and is NP-complete otherwise. This paper provides two new characterizations of finitely generated Taylor varieties. The first characterization is using absorbing subalgebras and the second one cyclic terms. These new conditions allow us to reprove the conjecture of Bang-Jensen and Hell (proved by the authors) and the characterization of locally finite Taylor varieties using weak near-unanimity terms (proved by McKenzie and Mar\'oti) in an elementary and self-contained way

Hereditarily hard H-colouring problems

AbstractLet 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

Cores and Compactness of Infinite Directed Graphs

AbstractIn this paper we define the property of homomorphic compactness for digraphs. We prove that if a digraphHis homomorphically compact thenHhas a core, although the converse does not hold. We also examine a weakened compactness condition and show that when this condition is assumed, compactness is equivalent to containing a core. We use this result to prove that if a digraphHof sizeκis not compact, then there is a digraphGof size at mostκ+such thatHis not compact with respect toG. We then give examples of some sufficient conditions for compactness

A graph theoretic proof of the complexity of colouring by a local tournament with at least two directed cycles

In this paper we give a graph theoretic proof of the fact that deciding whether a homomorphism exists to a fixed local tournament with at least two directed cycles is NP-complete. One of the main reasons for the graph theoretic proof is that it showcases all of the techniques that have been built up over the years in the study of the digraph homomorphism problem

A graph theoretic proof of the complexity of colouring by a local tournament with at least two directed cycles

In this paper we give a graph theoretic proof of the fact that deciding whether a homomorphism exists to a fixed local tournament with at least two directed cycles is NP-complete. One of the main reasons for the graph theoretic proof is that it showcases all of the techniques that have been built up over the years in the study of the digraph homomorphism problem

Graph Relations and Constrained Homomorphism Partial Orders

We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives from relations between graphs and is related to multihomomorphisms. This gives a generalization of surjective homomorphisms and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time. The theory of the graph homomorphism order is well developed, and from it we consider analogous notions defined for orders induced by constrained homomorphisms. We identify corresponding cores, prove or disprove universality, characterize gaps and dualities. We give a new and significantly easier proof of the universality of the homomorphism order by showing that even the class of oriented cycles is universal. We provide a systematic approach to simplify the proofs of several earlier results in this area. We explore in greater detail locally injective homomorphisms on connected graphs, characterize gaps and show universality. We also prove that for every $d\geq 3$ the homomorphism order on the class of line graphs of graphs with maximum degree $d$ is universal