Counting Homomorphisms to Trees Modulo a Prime

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 # p H OMS T O H , 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 that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of H on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree H and every prime p the problem # p H OMS T O H is either polynomial time computable or # p P-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of # p H OMS T O H are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime p . These results are the first suggesting that such dichotomies hold not only for the modulo 2 case but also for the modular counting functions of all primes p .

Solving Graph Homomorphism and Subgraph Isomorphism Problems Faster Through Clique Neighbourhood Constraints

Graph homomorphism problems involve finding adjacency-preserving mappings between two given graphs. Although theoretically hard, these problems can often be solved in practice using constraint programming algorithms. We show how techniques from the state-of-the-art in subgraph isomorphism solving can be applied to broader graph homomorphism problems, and introduce a new form of filtering based upon clique-finding. We demonstrate empirically that this filtering is effective for the locally injective graph homomorphism and subgraph isomorphism problems, and gives the first practical constraint programming approach to finding general graph homomorphisms.

Perfect matchings, rank of connection tensors and graph homomorphisms

We develop a theory of graph algebras over general fields. This is modelled after the theory developed by Freedman et al. (2007, J. Amer. Math. Soc.20 37–51) for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions with both complex vertex and edge weights (or even from more general fields). Our necessary and sufficient condition in terms of connection tensors is a simple exponential rank bound. It shows that positive semidefiniteness is not needed in the more general setting.

On a Theorem of Lovász that (&sdot, H ) Determines the Isomorphism Type of H

Graph homomorphism has been an important research topic since its introduction [20]. Stated in the language of binary relational structures in that paper [20], Lovász proved a fundamental theorem that, for a graph H given by its 0-1 valued adjacency matrix, the graph homomorphism function G ↦ hom( G , H ) determines the isomorphism type of H . In the past 50 years, various extensions have been proved by many researchers [1, 15, 21, 24, 26]. These extend the basic 0-1 case to admit vertex and edge weights; but these extensions all have some restrictions such as all vertex weights must be positive. In this article, we prove a general form of this theorem where H can have arbitrary vertex and edge weights. A noteworthy aspect is that we prove this by a surprisingly simple and unified argument. This bypasses various technical obstacles and unifies and extends all previous known versions of this theorem on graphs. The constructive proof of our theorem can be used to make various complexity dichotomy theorems for graph homomorphism effective in the following sense: it provides an algorithm that for any H either outputs a P-time algorithm solving hom(&sdot, H ) or a P-time reduction from a canonical #P-hard problem to hom(&sdot, H ).

The Creation of Network Intrusion Fingerprints by Graph Homomorphism

Attack attribution in cyber-attacks tends to be a qualitative exercise with a substantial room forerror. Graph theory is already a proven tool for modeling any connected system. Utilizing graph theory canprovide a quantitative, mathematically rigorous methodology for attack attribution. By identifyinghomomorphic subgraphs as points of comparison, one can create a fingerprint of an attack. That would allowone to match that fingerprint to new attacks and determine if the same threat actor conducted the attack. Thiscurrent study provides a mathematical method to create network intrusion fingerprints by applying graph theoryhomomorphisms. This provides a rigorous method for attack attribution. A case study is used to test thismethodology and determine its efficacy in identifying attacks perpetrated by the same threat actor and/or usingthe same threat vector.

Homomorphisms of Cayley graphs and Cycle Double Covers

We study the following conjecture of Matt DeVos: If there is a graph homomorphism from a Cayley graph $\mathrm{Cay}(M, B)$ to another Cayley graph $\mathrm{Cay}(M', B')$ then every graph with an $(M,B)$-flow has an $(M',B')$-flow. This conjecture was originally motivated by the flow-tension duality. We show that a natural strengthening of this conjecture does not hold in all cases but we conjecture that it still holds for an interesting subclass of them and we prove a partial result in this direction. We also show that the original conjecture implies the existence of an oriented cycle double cover with a small number of cycles.