We present a notion of convergence for sequences of finite graphs {Gn} that can be seen as a generalization of the Benjamini-Schramm convergence notion for bounded degree graphs, regarding the distribution of r-neighbourhoods of the vertices, and the left-convergence notion for dense graphs, regarding, given any finite graph F , the limit of the probabilities that a random map from V (F ) to V ...

-This paper is concerned with programming of the Potts mean field theory neural networks for pattern recognition by homomorphic mapping of the attributed relational graphs (ARG). In order to generate the homomorphic mapping from the scene relational graph to the model graph, we make use of the recently introduced [Suganthan, Technical Report, Nanyang Technical University (1994)] compatibility f...

A graph homomorphism from a graph G to a graph H is a mapping h : V (G)→ V (H) such that h(u) ∼ h(v) if u ∼ v. Graph homomorphisms are well studied objects and, for suitable choices of eitherG orH, many classical graph properties can be formulated in terms of homomorphisms. For example the question of wether there exists a homomorphism from G to H = Kq is the same as asking wether G is q-colour...

We study the complexity of the problem MAX SOL which is a natural optimisation version of the graph homomorphism problem. Given a fixed target graph H with V (H) ⊆ N, and a weight function w : V (G) → Q, an instance of the problem is a graph G and the goal is to find a homomorphism f : G → H which maximises P v∈G f(v) · w(v). MAX SOL can be seen as a restriction of the MIN HOM-problem [Gutin et...

Oriented chromatic number of an oriented graph G is the minimum order of an oriented graph H such that G admits a homomorphism to H . The oriented chromatic number of an unoriented graph G is the maximal chromatic number over all possible orientations of G. In this paper, we prove that every Halin graph has oriented chromatic number at most 8, improving a previous bound by Hosseini Dolama and S...

Correspondence homomorphisms are both a generalization of standard homomorphisms and a generalization of correspondence colourings. For a fixed target graph H, the problem is to decide whether an input graph G, with each edge labeled by a pair of permutations of V (H), admits a homomorphism to H ’corresponding’ to the labels, in a sense explained below. We classify the complexity of this proble...

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). If, moreover, each vertex u ∈ V (D) is associated with costs ci (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 (abbreviated MinHOM(H )). The problem is to decide, for an ...

In this paper we propose right-angled Artin groups as platform for a secret sharing scheme based on the efficiency (linear time) of the word problem. We define two new problems: subgroup isomorphism problem for Artin subgroups and group homomorphism problem in right-angled Artin groups. We show that the group homomorphism and graph homomorphism problems are equivalent, and the later is known to...

Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of #pHomsToH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p. Dyer and Greenhill proved a dichotomy stating t...

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 ci(u), i ∈ V (H), then the cost of a homomorphism f is ∑ u∈V (G) cf(u)(u). For each fixed digraph H, the minimum cost homomorphism problem for H, denoted MinHOM(H), can be formulated as follows: Given an input digra...