For a fixed graph $H$, by Hom($H$) we denote the computational problem which asks whether given $G$ admits homomorphism to i.e., an edge-preserving mapping from $V(G)$ $V(H)$. As Hom($K_k$) is equivalent $k$-Coloring, homomorphisms can be seen as generalizations of colorings. It known that polynomial-time solvable if $H$ bipartite or has vertex with loop, and NP-complete otherwise [Hell Nešetři...

In a nutshell, a duality for a constraint satisfaction problem equates the existence of one homomorphism to the non-existence of other homomorphisms. In this survey paper, we give an overview of logical, combinatorial, and algebraic aspects of the following forms of duality for constraint satisfaction problems: finite duality, bounded pathwidth duality, and bounded treewidth duality.

We study restricted homomorphism dualities in the context of classes with bounded expansion. This presents a generalization of restricted dualities obtained earlier for bounded degree graphs and also for proper minor closed classes. This is related to distance coloring of graphs and to the “approximative version” of Hadwiger conjecture.

Let A be a unital amenable A∗-algebra, or a unital commutative hermitian Banach ∗-algebra. We show that if ρ : A → B(H) is a bounded homomorphism, then ρ is completely bounded and ‖ρ‖cb ≤ ‖ρ‖2. Mathematics Subject Classification: 46L07; 46k05

Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This universal order is a distributive lattice. We investigate some other properties, namely the infinite distributivity, the computation of infinite suprema and...

We show that every interval in the homomorphism order of finite undirected graphs is either universal or a gap. Together with density and universality this “fractal” property contributes to the spectacular properties of the homomorphism order. We first show the fractal property by using Sparse Incomparability Lemma and then by a more involved elementary argument.

A homomorphism of an oriented graph G = (V,A) to an oriented graph G = (V , A) is a mapping φ from V to V ′ such that φ(u)φ(v) is an arc in G whenever uv is an arc in G. A homomorphism of G to G is said to be T -preserving for some oriented graph T if for every connected subgraph H of G isomorphic to a subgraph of T , H is isomorphic to its homomorphic image in G. The T -preserving oriented chr...

Denote by G and D, respectively, the the homomorphism poset of the finite undirected and directed graphs, respectively. A maximal antichain A in a poset P splits if A has a partition (B, C) such that for each p ∈ P either b ≤P p for some b ∈ B or p ≤p c for some c ∈ C. We construct both splitting and non-splitting infinite maximal antichains in G and in D. A point y ∈ P is a cut point in a pose...