Related papers: Alternative parameterizations of Metric Dimension

The Metric Dimension problem asks for a minimum-sized resolving set in a given (unweighted, undirected) graph $G$. Here, a set $S \subseteq V(G)$ is resolving if no two distinct vertices of $G$ have the same distance vector to $S$. The…

A set of vertices $W$ {\em resolves} a graph $G$ if every vertex of $G$ is uniquely determined by its vector of distances to the vertices in $W$. The {\em metric dimension} for $G$, denoted by $\dim(G)$, is the minimum cardinality of 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…

The metric dimension of a graph $G$ is the size of a smallest subset $L \subseteq V(G)$ such that for any $x,y \in V(G)$ with $x\not= y$ there is a $z \in L$ such that the graph distance between $x$ and $z$ differs from the graph distance…

Let $G=(V,E)$ be a connected simple graph. The distance $d(u,v)$ between vertices $u$ and $v$ from $V$ is the number of edges in the shortest $u-v$ path. If $e=uv \in E$ is an edge in $G$ than distance $d(w,e)$ where $w$ is some vertex in…

The notion of resolving sets in a graph was introduced by Slater (1975) and Harary and Melter (1976) as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices x and…

Let $G$ be a connected graph and $d(a,b)$ be the distance between the vertices $a$ and $b$. A subset $U =\{u_1,u_2,\cdots,u_k\}$ of the vertices is called a resolving set for $G$ if for every two distinct vertices $a,b \in V(G)$, there is a…

The metric dimension, $\dim(G)$, of a graph $G$ is a graph parameter motivated by robot navigation that has been studied extensively. Let $G$ be a graph with vertex set $V(G)$, and let $d(x,y)$ denote the length of a shortest $x-y$ path in…

Let $G$ be a connected graph. Given an ordered set $W = \{w_1, w_2,\dots w_k\}\subseteq V(G)$ and a vertex $u\in V(G)$, the representation of $u$ with respect to $W$ is the ordered $k$-tuple $(d(u,w_1), d(u,w_2),\dots,$ $d(u,w_k))$, where…

An ordered set $S$ of vertices of a graph $G$ is a resolving set for $G$ 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. In…

Let $G=(V,E)$ be a connected graph and $d_{G}(u,v)$ be the shortest distance between the vertices $u$ and $v$ in $G$. A set $S=\{s_{1},s_{2},\cdots,s_{n}\}\subset V(G)$ is said to be a {\em resolving set} if for all distinct vertices $u,v$…

Let $G=(V,E)$ be a connected graph. A vertex $w\in V$ distinguishes two elements (vertices or edges) $x,y\in E\cup V$ if $d_G(w,x)\ne d_G(w,y)$. A set $S$ of vertices in a connected graph $G$ is a mixed metric generator for $G$ if every two…

Let $\Gamma$ be a simple connected undirected graph with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$. The metric dimension of a graph $\Gamma$ is the least number of vertices in a set with the property that the list of distances from…

A vertex $v\in V$ is said to resolve two vertices $x$ and $y$ if $d_G(v,x)\ne d_G(v,y)$. A set $S\subset V$ is said to be a metric generator for $G$ if any pair of vertices of $G$ is resolved by some element of $S$. A minimum metric…

A vertex set $U \subseteq V$ of an undirected graph $G=(V,E)$ is a \textit{resolving set} for $G$ if for every two distinct vertices $u,v \in V$ there is a vertex $w \in U$ such that the distance between $u$ and $w$ and the distance between…

For a graph $G = (V,E)$ and a subset $R \subseteq V$, we say that $R$ is \textit{multiset resolving} for $G$ if for every pair of vertices $v,w$, the \textit{multisets} $\{d(v,r): r \in R\}$ and $\{d(w,r):r \in R\}$ are distinct, where…

A set $R \subseteq V(G)$ is a resolving set of a graph $G$ if for all distinct vertices $v,u \in V(G)$ there exists an element $r \in R$ such that $d(r,v) \neq d(r,u)$. The metric dimension $\dim(G)$ of the graph $G$ is the minimum…

Let $G$ be a strongly connected directed graph and $u,v,w\in V(G)$ be three vertices. Then $w$ strongly resolves $u$ to $v$ if there is a shortest $u$-$w$-path containing $v$ or a shortest $w$-$v$-path containing $u$. A set $R\subseteq…

Let $G$ be a connected graph. A vertex $w$ strongly resolves a pair $u$, $v$ of vertices of $G$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $W$ of vertices is a strong resolving…