Related papers: Edge Quasi $\lambda$-distance-balanced Graphs in M…
The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…
Due to their capacity to encode rich structural information, labeled graphs are often used for modeling various kinds of objects such as images, molecules, and chemical compounds. If pattern recognition problems such as clustering and…
A monitoring edge-geodetic set, or simply an MEG-set, of a graph $G$ is a vertex subset $M \subseteq V(G)$ such that given any edge $e$ of $G$, $e$ lies on every shortest $u$-$v$ path of $G$, for some $u,v \in M$. The monitoring…
A monitoring edge-geodetic set (or meg-set for short) of a graph is a set of vertices $M$ such that if any edge is removed, then the distance between some two vertices of $M$ increases. This notion was introduced by Foucaud et al. in 2023…
A signed graph is one that features two types of edges: positive and negative. Balanced signed graphs are those in which all cycles contain an even number of positive edges. In the adjacency matrix of a signed graph, entries can be $0$,…
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and $f$ be a 0-1 labeling of $E(G)$ so that the absolute difference in the number of edges labeled 1 and 0 is no more than one. Call such a labeling $f$ \emph{edge-friendly}. We…
Computing efficiently a robust measure of similarity or dissimilarity between graphs is a major challenge in Pattern Recognition. The Graph Edit Distance (GED) is a flexible measure of dissimilarity between graphs which arises in…
Predicting relaxed atomic structures of chemically complex materials remains a major computational challenge, particularly for high-entropy systems where traditional first-principles methods become prohibitively expensive. We introduce the…
Statistical graph models aim at modeling graphs as random realization among a set of possible graphs. One issue is to evaluate whether or not a graph is likely to have been generated by one particular model. In this paper we introduce the…
A graph G is {\xi}-nearly planar if it can be embedded in the sphere so that each of its edges is crossed at most {\xi} times. The family of {\xi}-nearly planar graphs is widely extending the notion of planarity. We introduce an alternative…
Quasi-isometry is a measure of how similar two graphs are at `large-scale'. Nguyen, Scott, and Seymour [arXiv:2501.09839] and Hickingbotham [arXiv:2501.10840] independently gave a characterisation of graphs quasi-isometric to graphs of…
For a given graph $G$, the metric and edge metric dimensions of $G$, $\dim(G)$ and ${\rm edim}(G)$, are the cardinalities of the smallest possible subsets of vertices in $V(G)$ such that they uniquely identify the vertices and the edges of…
For a compact set $E \subset \mathbb R^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k+1$ points in $E$ such that the distance between a pair of points is specified if the corresponding…
The metric (resp. edge metric or mixed metric) dimension of a graph $G$, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of $G$ by…
The edge-Szeged index of a graph $G$ is defined as $Sz_e(G) = \sum_{e=uv \in E(G)}m_u(e)m_v(e)$, where $m_u(e)$ denotes the number of edges of $G$ whose distance to $u$ is smaller than the distance to $v$ and $m_v(e)$ denotes the number of…
Graph Sampling provides an efficient yet inexpensive solution for analyzing large graphs. While extracting small representative subgraphs from large graphs, the challenge is to capture the properties of the original graph. Several sampling…
Dominating sets and resolving sets have important applications in control theory and computer science. In this paper, we introduce an edge-analog of the classical dominant metric dimension of graphs. By combining the concepts of a…
For arbitrary semimetric space $(X, d)$ and disjoint proximinal subsets $A$, $B$ of $X$ we define the proximinal graph as a bipartite graph with parts $A$ and $B$ whose edges $\{a, b\}$ satisfy the equality $d(a, b) = \operatorname{dist}(A,…
The concept of a monitoring edge-geodetic set (MEG-set) in a graph $G$, denoted $MEG(G)$, refers to a subset of vertices $MEG(G)\subseteq V(G)$ such that every edge $e$ in $G$ is monitored by some pair of vertices $ u, v \in MEG(G)$, where…
A connected graph $\Gamma$ of diameter ${\rm diam}(\Gamma) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}(\Gamma)|=|W_{yx}(\Gamma)|$ for every $x,y\in V(\Gamma)$ with $d_{\Gamma}(x,y)=\ell$, where $W_{xy}(\Gamma)$ is the set of vertices…