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Recently, Man\v{c}inska and Roberson proved that two graphs $G$ and $G'$ are quantum isomorphic if and only if they admit the same number of homomorphisms from all planar graphs. We extend this result to planar #CSP with any pair of sets…
It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…
An observable canonical form is formulated for the set of rational systems on a variety each of which is a single-input-single-output, affine in the input, and a minimal realization of its response map. The equivalence relation for the…
We give an algorithm that, for every fixed k, decides isomorphism of graphs of rank width at most k in polynomial time. As the clique width of a graph is bounded in terms of its rank width, we also obtain a polynomial time isomorphism test…
Graph isomorphism is an important computer science problem. The problem for the general case is unknown to be in polynomial time. The base algorithm for the general case works in quasi-polynomial time. The solutions in polynomial time for…
In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some…
Homomorphism indistinguishability is a way of characterising many natural equivalence relations on graphs. Two graphs $G$ and $H$ are called homomorphism indistinguishable over a graph class $\mathcal{F}$ if for each $F \in \mathcal{F}$,…
Symmetries and isomorphisms play similar conceptual roles when we consider how models represent physical situations, but they are formally distinct, as two models related by symmetries are not typically isomorphic. I offer a rigorous…
We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in…
Let us be given two graphs $\Gamma_1$, $\Gamma_2$ of $n$ vertices. Are they isomorphic? If they are, the set of isomorphisms from $\Gamma_1$ to $\Gamma_2$ can be identified with a coset $H\cdot\pi$ inside the symmetric group on $n$…
In the $(G,H)$-isomorphism game, a verifier interacts with two non-communicating players (called provers) by privately sending each of them a random vertex from either $G$ or $H$, whose aim is to convince the verifier that two graphs $G$…
The homomorphism problem for relational structures is an abstract way of formulating constraint satisfaction problems (CSP) and various problems in database theory. The decision version of the homomorphism problem received a lot of…
Let $G$ be an undirected graph on $n$ vertices and let $S(G)$ be the set of all $n \times n$ real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of $G$. The inverse eigenvalue…
We propose a simple and efficient local algorithm for graph isomorphism which succeeds for a large class of sparse graphs. This algorithm produces a low-depth canonical labeling, which is a labeling of the vertices of the graph that…
The search for a highly discriminating and easily computable invariant to distinguish graphs remains a challenging research topic. Here we focus on cospectral graphs whose complements are also cospectral (generalized cospectral), and on…
An explicit algorithm is presented for testing whether two non-directed graphs are isomorphic or not. It is shown that for a graph of n vertices, the number of n independent operations needed for the test is polynomial in n. A proof that…
The graph isomorphism problem is considered. We assign modified $n$-variable characteristic polynomials for graphs and reduce the graph isomorphism problem to the problem of the polynomials isomorphism. It is required to find out, is there…
We prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colorings, edge colorings and/or edge…
Graph isomorphism is an important problem as its worst-case time complexity is not yet fully understood. In this study, we try to draw parallels between a related optimization problem called point set registration. A graph can be…
We study reconfiguration problems for cliques in a graph, which determine whether there exists a sequence of cliques that transforms a given clique into another one in a step-by-step fashion. As one step of a transformation, we consider…