We fully characterise the situations where the existence of a homomorphism from a digraph G to at least one of a finite set H of directed graphs is determined by a finite number of forbidden subgraphs. We prove that these situations, called generalised dualities, are characterised by the non-existence of a homomorphism to G from a finite set of forests. Furthermore, we characterise all finite m...

In this talk we will discuss some examples of graph homomorphisms. More precisely, the graph parameters which can be represented by counting the graph homomorphisms. The main reference is Section 2 in [2].

For a graph G, let     max : is an edge cut of b G D D G  . For graphs G and H, a map     :V G V H   is a graph homomorphism if for each   e uv E G   ,       u v E H    . In 1979, Erdös proved by probabilistic methods that for p ≥ 2 with

A graph homomorphism is a vertex map which carries edges from a source graph to edgesin a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem(MAX H -COL) are edge-weighted graphs G and the objective is to find a subgraph of G that hasmaximal total edge weight, under the condition that the subgraph has a homomorphism to H ;note that for H = Kk th...

The semiring-based constraint satisfaction problems (semiring CSPs), proposed by Bistarelli, Montanari and Rossi [3], is a very general framework of soft constraints. In this paper we propose an abstraction scheme for soft constraints that uses semiring homomorphism. To find optimal solutions of the concrete problem, the idea is, first working in the abstract problem and finding its optimal sol...

Given a countable Borel equivalence relation E on a Polish space, let IE denote the σ-ideal generated by the Borel partial transversals of E. We show that there is a Borel homomorphism from IE to IF if and only if there is a smooth-to-one Borel homomorphism from a finite index Borel subequivalence relation of E to F . As a corollary, we see that IE is homogeneous in the sense of Zapletal (2007,...

Let D denote the partially ordered sets of homomorphism classes of finite directed graphs, ordered by the homomorphism relation. Order theoretic properties of this poset have been studied extensively, and have interesting connections to familiar graph properties and parameters. This paper studies the generalized duality pairs in D: it gives a new, short proof for the Foniok Nešetřil Tardif theo...