Related papers: Edge metric dimension of some graph operations

A graph is $k$-chordal if it does not have an induced cycle with length greater than $k$. We call a graph chordal if it is $3$-chordal. Let $G$ be a graph. The distance between the vertices $x$ and $y$, denoted by $d_{G}(x,y)$, is the…

The direct product of graphs $G=(V(G),E(G))$ and $H=(V(H),E(H))$ is the graph, denoted as $G\times H$, with vertex set $V(G\times H)=V(G)\times V(H)$, where vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent in $G\times H$ if $x_1x_2\in…

A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}$ such…

Foucaud et al. [Discrete Appl. Math. 319 (2022), 424-438] recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. For a set $M$ of vertices and an edge $e$ of a graph $G$, let $P(M,…

Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut dimension of $G$ is the dimension of the span of the characteristic vectors of the minimum cuts of $G$, viewed as vectors in $\{0,1\}^m$. For every $n \ge 2$ we show…

In this paper, we introduce a connection between two classical concepts of graph theory: \; metric dimension and distinguishing number. For a given graph $G$, let ${\rm dim}(G)$ and $D(G)$ represent its metric dimension and distinguishing…

Erd\H{o}s, Harary, and Tutte defined the dimension of a graph $G$ as the smallest natural number $n$ such that $G$ can be embedded in $\mathbb{R}^n$ with each edge a straight line segment of length 1. Since the proposal of this definition,…

A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$ if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from $u$ than $v$. The boundary $\partial(G)$ of $G$ is the set of all of its boundary…

Let $G$ be a connected finite graph with vertex set $V(G)$. The eccentricity $e(v)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is defined as $\frac{1}{|V(G)|}\sum_{v \in…

This article investigates the connectivity dimension of a graph. We introduce this concept in analogy to the metric dimension of a graph, providing a graph parameter that measures the heterogeneity of the connectivity structure of a graph.…

A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the…

In this paper we offer a metric similar to graph edit distance which measures the distance between two (possibly infinite)weighted graphs with finite norm (we define the norm of a graph as the sum of absolute values of its edges). The main…

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the…

We introduce a new graph-theoretic concept in the area of network monitoring. A set $M$ of vertices of a graph $G$ is a \emph{distance-edge-monitoring set} if for every edge $e$ of $G$, there is a vertex $x$ of $M$ and a vertex $y$ of $G$…

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner…

Foucaud {\it et al.} recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Let $G$ be a graph with vertex set $V(G)$, $M$ a subset of $V(G)$, and $e$ be an edge in $E(G)$, and let…

Let $G$ be a graph with vertex set $V(G)$. For any two distinct vertices $x$ and $y$ of $G$, let $R\{x, y\}$ denote the set of vertices $z$ such that the distance from $x$ to $z$ is not equal to the distance from $y$ to $z$ in $G$. For a…

The metric dimension of a graph measures how uniquely vertices may be identified using a set of landmark vertices. This concept is frequently used in the study of network architecture, location-based problems and communication. Given a…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

The edge-connectivity matrix of a weighted graph is the matrix whose off-diagonal $v$-$w$ entry is the weight of a minimum edge cut separating vertices $v$ and $w$. Its computation is a classical topic of combinatorial optimization since at…