Related papers: The Graph Isomorphism Problem and approximate cate…
In recent years, algorithms and neural architectures based on the Weisfeiler--Leman algorithm, a well-known heuristic for the graph isomorphism problem, have emerged as a powerful tool for machine learning with graphs and relational data.…
Two graphs $G$ and $H$ are homomorphism indistinguishable over a family of graphs $\mathcal{F}$ if for all graphs $F \in \mathcal{F}$ the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphism from $F$ to $H$. Many…
We examine ordered graphs, defined as graphs with linearly ordered vertices, from the perspective of homomorphisms (and colorings) and their complexities. We demonstrate the corresponding computational and parameterized complexities, along…
Seminal research in the field of graph neural networks (GNNs) has revealed a direct correspondence between the expressive capabilities of GNNs and the $k$-dimensional Weisfeiler-Leman ($k$WL) test, a widely-recognized method for verifying…
Current GNN architectures use a vertex neighborhood aggregation scheme, which limits their discriminative power to that of the 1-dimensional Weisfeiler-Lehman (WL) graph isomorphism test. Here, we propose a novel graph convolution operator…
The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general…
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…
We consider the following two algorithmic problems: given a graph $G$ and a subgraph $H\subseteq G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved…
The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width are known to be solvable in polynomial time [Bod90],[YBFT99]. We give restricted space algorithms for these problems proving the…
We relate the graph isomorphism problem to the solvability of certain systems of linear equations with nonnegative variables. This version replaces the two previous versions of this paper.
There is no known polynomial-time algorithm for graph isomorphism testing, but elementary combinatorial "refinement" algorithms seem to be very efficient in practice. Some philosophical justification is provided by a classical theorem of…
We construct a polynomial-time algorithm which given a graph $\Gamma$ finds the full set of non-equivalent Cayley representations of $\Gamma$ over the group $D\cong C_p\times C_{p^k}$, where $p\in\{2,3\}$ and $k\geq 1$. This result implies…
Let $G$ be a group. The BCI problem asks whether two Haar graphs of $G$ are isomorphic if and only if they are isomorphic by an element of an explicit list of isomorphisms. We first generalize this problem in a natural way and give a…
Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two…
An isomorphism between two graphs is a bijection between their vertices that preserves the edges. We consider the problem of determining whether two finite undirected weighted graphs are isomorphic, and finding an isomorphism relating them…
The criteria for determining graph isomorphism are crucial for solving graph isomorphism problems. The necessary condition is that two isomorphic graphs possess invariants, but their function can only be used to filtrate and subdivide…
Line graph transformation has been widely studied in graph theory, where each node in a line graph corresponds to an edge in the original graph. This has inspired a series of graph neural networks (GNNs) applied to transformed line graphs,…
The individualization-refinement paradigm provides a strong toolbox for testing isomorphism of two graphs and indeed, the currently fastest implementations of isomorphism solvers all follow this approach. While these solvers are fast in…
The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical computer science. For graphs $G$ and $H$, it asks whether they are the same up to a relabeling of vertices. In 1981, Lubiw proved that list…
The group isomorphism problem asks whether two finite groups given by their Cayley tables are isomorphic or not. Although there are polynomial-time algorithms for some specific group classes, the best known algorithm for testing isomorphism…