Related papers: Graph Isomorphism and Circuit Size
Problems related to graph matching and isomorphisms are very important both from a theoretical and practical perspective, with applications ranging from image and video analysis to biological and biomedical problems. The graph matching…
The state-of-the-art solvers for the graph isomorphism problem can readily solve generic instances with tens of thousands of vertices. Indeed, experiments show that on inputs without particular combinatorial structure the algorithms scale…
By a map we mean a $2$-cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. Automorphism of a map can be thought of as a permutation of the vertices which…
We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism $\phi$ between two isomorphic graphs is as hard as computing $\phi$ itself. This result optimally improves upon a result of G\'{a}l et al.…
As it is well known, the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-F\"urer-Immerman…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers where this homomorphism is prescribed by the action of a semiregular subgroup of $\textrm{Aut}(G)$. We study…
In this paper we are interested in the fine-grained complexity of deciding whether there is a homomorphism from an input graph $G$ to a fixed graph $H$ (the $H$-Coloring problem). The starting point is that these problems can be viewed as…
The problem of graph isomorphism is an important but challenging problem in the field of graph analysis, for example: analyzing the similarity of two chemical molecules, or studying the expressive ability of graph neural networks. WL test…
The graph reconstruction conjecture states that all graphs on at least three vertices are determined up to isomorphism by their deck. In this paper, a general framework for this problem is proposed to simply explain the reconstruction of…
We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in $\Sigma_2^p$. In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is…
Graph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H. Given H it is easy to compute its square H2, however Motwani and Sudan proved that it is NP-complete to determine…
While graphs and abstract data structures can be large and complex, practical instances are often regular or highly structured. If the instance has sufficient structure, we might hope to compress the object into a more succinct…
There is a graph reduction system so that every optimal 1-planar graph can be reduced to an irreducible extended wheel graph, provided the reductions are applied such that the given graph class is preserved. A graph is optimal 1-planar if…
Graph embedding, especially as a subgraph of a grid, is an old topic in VLSI design and graph drawing. In this paper, we investigate related questions concerning the complexity of embedding a graph $G$ in a host graph that is the strong…
We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges…
A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of ${\rm Aut}(G)$. Regular…
In recent years there has been a rapid increase in classification methods on graph structured data. Both in graph kernels and graph neural networks, one of the implicit assumptions of successful state-of-the-art models was that…
In this paper we study the following problem. Let $A$ be a fixed graph, and let $\hom(G,A)$ denote the number of homomorphisms from a graph $G$ to $A$. Furthermore, let $v(G)$ denote the number of vertices of $G$, and let $\mathcal{G}_d$…
Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of…