In 2004, Berdine, Calcagno and O’Hearn introduced a fragment of separation logic that allows for reasoning about programs with pointers and linked lists. They showed that entailment in this fragment is in coNP, but the precise complexity of this problem has been open since. In this paper, we show that the problem can actually be solved in polynomial time. To this end, we represent separation lo...

A signed graph is a together with an assignment of signs to the edges. closed walk in said be positive (negative) if it has even (odd) number negative edges, counting repetition. Recognizing walks as one key structural properties graph, we define homomorphism (G,?) (H,?) mapping vertices and edges G (respectively) H which preserves incidence, adjacency walks. In this work first give characteriz...

Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such partition functions in statistical physics, particular independent sets and colourings. In this article, we study the complexity of # p H OMS T O , problem homomorphisms from an input to a modulo prime number . Dyer Greenhill proved dichotomy stating that tractability non-modular depends on struc...

A homomorphism from a graph G to a graph H is locally bijective, surjective, or injective if its restriction to the neighborhood of every vertex of G is bijective, surjective, or injective, respectively. We prove that the problems of testing whether a given graph G allows a homomorphism to a given graph H that is locally bijective, surjective, or injective, respectively, are NP-complete, even w...

A Theta graph is a multigraph which is a union of at least three internally disjoint paths that have the same two distinct end vertices. We show full computational complexity characterization of the problem of deciding the existence of a locally injective homomorphism from an input graph G to any fixed Theta graph.

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) cf(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H. The problem is to decide, for an input graph D with costs ci(u), u...

Simple Conceptual Graphs (SGs) are used to represent entities and relations between these entities: they can be translated into positive, conjunctive, existential first-order logics, without function symbols. Sound and complete reasonings w.r.t. associated logic formulas are obtained through a kind of graph homomorphism called projection. Conceptual Graphs Rules (or CG rules) are a standard ext...

Connection matrices were introduced in [1], where they were used to characterize graph homomorphism functions. The goal of this note is to determine the exact rank of these matrices. The result can be rephrased in terms of graph algebras, also introduced in [1]. Yet another version proves that if two k-tuples of nodes behave the same way from the point of view of graph homomorphisms, then they ...

A result of Fiz Pontiveros shows that if A is a random subset of ZN where each element is chosen independently with probability N−1/2+o(1), then with high probability every Freiman homomorphism defined on A can be extended to a Freiman homomorphism on the whole of ZN . In this paper we improve the bound to CN−2/3(logN)1/3, which is best possible up to the constant factor.

Tension-continuous (shortly TT ) mappings are mappings between the edge sets of graphs. They generalize graph homomorphisms. From another perspective, tension-continuous mappings are dual to the notion of flow-continuous mappings and the context of nowhere-zero flows motivates several questions considered in this paper. Extending our earlier research we define new constructions and operations f...