Related papers: Towards an Isomorphism Dichotomy for Hereditary Gr…
For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring…
Subgraph and homomorphism counting are fundamental algorithmic problems. Given a constant-sized pattern graph $H$ and a large input graph $G$, we wish to count the number of $H$-homomorphisms/subgraphs in $G$. Given the massive sizes of…
Graphs constructed to translate some graph problem into another graph problem are usually called auxiliary graphs. Specifically total graphs of simple graphs are used to translate the total colouring problem of the original graph into a…
The computational complexity of the Vertex Coloring problem is known for all hereditary classes of graphs defined by forbidding two connected five-vertex induced subgraphs, except for seven cases. We prove the polynomial-time solvability of…
It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many…
We introduce two novel evolutionary formulations of the problem of coloring the nodes of a graph. The first formulation is based on the relationship that exists between a graph's chromatic number and its acyclic orientations. It views such…
We consider the $\#\mathsf{W}[1]$-hard problem of counting all matchings with exactly $k$ edges in a given input graph $G$; we prove that it remains $\#\mathsf{W}[1]$-hard on graphs $G$ that are line graphs or bipartite graphs with degree…
List colouring is an NP-complete decision problem even if the total number of colours is three. It is hard even on planar bipartite graphs. We give a polynomial-time algorithm for solving list colouring of permutation graphs with a bounded…
We develop the methodology of positioning graph vertices relative to each other to solve the problem of determining isomorphism of two undirected graphs. Based on the position of the vertex in one of the graphs, it is determined the…
For a class $\mathcal{G}$ of graphs, the objective of \textsc{Subgraph Complementation to} $\mathcal{G}$ is to find whether there exists a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph…
A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…
We show how graph neural networks can be used to solve the canonical graph coloring problem. We frame graph coloring as a multi-class node classification problem and utilize an unsupervised training strategy based on the statistical physics…
Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number $k$, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in $k$ steps only few vertices with…
We prove that the degeneracy of graphs in a hereditary class defined by a finite set S of forbidden induced subgraphs is bounded if and only if S includes a complete graph, a complete bipartite graph and a forest.
A coloured graph is k-ultrahomogeneous if every isomorphism between two induced subgraphs of order at most k extends to an automorphism. A coloured graph is t-tuple regular if the number of vertices adjacent to every vertex in a set S of…
The $H$-Coloring problem is a well-known generalization of the classical NP-complete problem $k$-Coloring where the task is to determine whether an input graph admits a homomorphism to the template graph $H$. This problem has been the…
Baker devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a…
We extend the concept of graph isomorphisms to multilayer networks with any number of "aspects" (i.e., types of layering). In developing this generalization, we identify multiple types of isomorphisms. For example, in multilayer networks…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdos conjectured that the random 2-edge-coloring minimizes the number of…