Related papers: Metric dimension for random graphs

Let $(X,d)$ be a metric space. A set $S\subseteq X$ is said to be a $k$-metric generator for $X$ if and only if for any pair of different points $u,v\in X$, there exist at least $k$ points $w_1,w_2, \ldots w_k\in S$ such that $d(u,w_i)\ne…

A resolving set of a graph is a set of vertices with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. In this paper, we construct a resolving set of Johnson graphs, doubled Odd…

Given a graph $G = (V,E)$, a set $S \subset V$ is called a $k$-\emph{metric generator} for $G$ if any pair of different vertices of $G$ is distinguished by at least $k$ elements of $S$. A graph is $k$-\emph{metric dimensional} if $k$ is the…

Let $G$ be a graph, and let $u$, $v$, and $w$ be vertices of $G$. If the distance between $u$ and $w$ does not equal the distance between $v$ and $w$, then $w$ is said to resolve $u$ and $v$. The metric dimension of $G$, denoted $\beta(G)$,…

The metric dimension dim(G) of a graph $G$ is the minimum cardinality of a subset $S$ of vertices of $G$ such that each vertex of $G$ is uniquely determined by its distances to $S$. It is well-known that the metric dimension of a graph can…

The classical (vertex) metric dimension of a graph G is defined as the cardinality of a smallest set S in V (G) such that any two vertices x and y from G have different distances to least one vertex from S: The k-metric dimension is a…

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric…

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),.,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where…

Let $G$ be a finite solvable group and let $\Delta(G)$ be the character degree graph of $G$. In this paper, we obtain the metric dimension of certain character degree graphs. Specifically, we calculate the metric dimension for a regular…

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 resolving set for a graph $\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension $\mu(\Gamma)$ is the smallest size…

As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if the elements of…

The outer multiset dimension ${\rm dim}_{\rm ms}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that…

Let $G(V, E)$ be a connected simple undirected graph. In this paper we prove that the edge metric dimension (introduced by Kelenc, Tratnik and Yero) of the Erd\H{o}s-R\'enyi random graph $G(n, p)$ is given by: $$\textrm{edim}(G(n, p)) = (1…

The unit ball random geometric graph $G=G^d_p(\lambda,n)$ has as its vertices $n$ points distributed independently and uniformly in the $d$-dimensional unit ball, with two vertices adjacent if and only if their $l_p$-distance is at most…

Let $G$ be a graph and $S\subseteq V(G)$. If every two adjacent vertices of $G$ have different metric $S$-representations, then $S$ is a local metric generator for $G$. A local metric generator of smallest order is a local metric basis for…

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…

For an ordered set W = {w1,w2,...,wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W) := (d(v,w1),d(v,w2),...,d(v,wk)) is called the (metric) representation of v with respect to W, where d(x,y) is the distance…

Let $G$ be a (di)graph. A set $W$ of vertices in $G$ is a \emph{resolving set} of $G$ if every vertex $u$ of $G$ is uniquely determined by its vector of distances to all the vertices in $W$. The \emph{metric dimension} $\mu (G)$ of $G$ is…

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.…