Related papers: Resolving sets tolerant to failures in three-dimen…

A vertex $w$ of a connected graph $G$ strongly resolves two vertices $u,v\in V(G)$, if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $S$ of vertices is a strong metric generator for…

Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its…

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…

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…

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…

A set of vertices $S\subseteq V(G)$ is a basis or resolving set of a graph $G$ if for each $x,y\in V(G)$ there is a vertex $u\in S$ such that $d(x,u)\neq d(y,u)$. A basis $S$ is a fault-tolerant basis if $S\setminus \{x\}$ is a basis for…

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number pi(G) so that every configuration…

A \emph{resolving set} $R$ in a graph $G$ is a set of vertices such that every vertex of $G$ is uniquely identified by its distances to the vertices of $R$. Introduced in the 1970s, this concept has been since then extensively studied from…

Given a set of vertices $S=\{v_1,v_2,...,v_k\}$ of a connected graph $G$, the metric representation of a vertex $v$ of $G$ with respect to $S$ is the vector $r(v|S)=(d(v,v_1),d(v,v_2),...,d(v,v_k))$, where $d(v,v_i)$, $i\in \{1,...,k\}$…

A locating-dominating set in a graph G is a subset of vertices representing "detectors" which can locate an "intruder" given that each detector covers its closed neighborhood and can distinguish its own location from its neighbors. We…

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…

Nonlocal metric dimension ${\rm dim}_{\rm n\ell}(G)$ of a graph $G$ is introduced as the cardinality of a smallest nonlocal resolving set, that is, a set of vertices which resolves each pair of non-adjacent vertices of $G$. Graphs $G$ with…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…

Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. For an ordered $k$-partition $\Pi=\{Q_1,\ldots,Q_k\}$ of $V(G)$, the representation of a vertex $v \in V(G)$ with respect to $\Pi$ is the $k$-vectors…

Let $G=(V,E)$ be a connected graph and let $d(u,v)$ denote the distance between vertices $u,v \in V$. A metric basis for $G$ is a set $B\subseteq V$ of minimum cardinality such that no two vertices of $G$ have the same distances to all…

A generator of a metric space is a set $S$ of points in the space with the property that every point of the space is uniquely determined by its distances from the elements of $S$. Given a simple graph $G=(V,E)$, we define the distance…

In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number gamma G is the minimum cardinality of a…

In this paper we study the complexity of the following problems: Given a colored graph X=(V,E,c), compute a minimum cardinality set S of vertices such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem…

A set $D$ of vertices of a graph $G$ is a dissociation set if each vertex of $D$ has at most one neighbor in $D$. The dissociation number of $G$, $diss(G)$, is the cardinality of a maximum dissociation set in a graph $G$. In this paper we…

We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the…