On the strong metric dimension of Cartesian sum graphs

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 $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $S$. The smallest cardinality of a strong metric generator for $G$ is called the strong metric dimension of $G$. In this paper we obtain several tight bounds or closed formulae for the strong metric dimension of the Cartesian sum of graphs in terms of the strong metric dimension, clique number or twins-free clique number of its factor graphs.Comment: arXiv admin note: text overlap with arXiv:1402.266

Computing the metric dimension of a graph from primary subgraphs

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 $d(u,w_i)$ denotes the distance between $u$ and $w_i$. The set $W$ is a metric generator for $G$ if every two different vertices of $G$ have distinct representations. A minimum cardinality metric generator is called a \emph{metric basis} of $G$ and its cardinality is called the \emph{metric dimension} of G. It is well known that the problem of finding the metric dimension of a graph is NP-Hard. In this paper we obtain closed formulae for the metric dimension of graphs with cut vertices. The main results are applied to specific constructions including rooted product graphs, corona product graphs, block graphs and chains of graphs.Comment: 18 page

Simultaneous Resolvability in Families of Corona Product Graphs

Let ${\cal G}$ be a graph family defined on a common vertex set $V$ and let $d$ be a distance defined on every graph $G\in {\cal G}$. A set $S\subset V$ is said to be a simultaneous metric generator for ${\cal G}$ if for every $G\in {\cal G}$ and every pair of different vertices $u,v\in V$ there exists $s\in S$ such that $d(s,u)\ne d(s,v)$. The simultaneous metric dimension of ${\cal G}$ is the smallest integer $k$ such that there is a simultaneous metric generator for ${\cal G}$ of cardinality $k$. We study the simultaneous metric dimension of families composed by corona product graphs. Specifically, we focus on the case of two particular distances defined on every $G\in {\cal G}$, namely, the geodesic distance $d_G$ and the distance $d_{G,2}:V\times V\rightarrow \mathbb{N}\cup \{0\}$ defined as $d_{G,2}(x,y)=\min\{d_{G}(x,y),2\}$.Comment: arXiv admin note: substantial text overlap with arXiv:1504.0049

The local metric dimension of the lexicographic product of graphs

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension and the local metric dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product $G \circ \mathcal{H}$ of a connected graph $G$ of order $n$ and a family $\mathcal{H}$ composed by $n$ graphs. We show that the local metric dimension of $G \circ \mathcal{H}$ can be expressed in terms of the true twin equivalence classes of $G$ and the local adjacency dimension of the graphs in $\mathcal{H}$

On the adjacency dimension of graphs

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 function $d_{G,2}:V\times V\rightarrow \mathbb{N}\cup \{0\}$, as $d_{G,2}(x,y)=\min\{d_G(x,y),2\},$ where $d_G(x,y)$ is the length of a shortest path between $x$ and $y$ and $\mathbb{N}$ is the set of positive integers. Then $(V,d_{G,2 })$ is a metric space. We say that a set $S\subseteq V$ is a $k$-adjacency generator for $G$ if for every two vertices $x,y\in V$, there exist at least $k$ vertices $w_1,w_2,...,w_k\in S$ such that d_{G,2}(x,w_i)\ne d_{G,2}(y,w_i),\; \mbox{for every}\; i\in \{1,...,k\}. A minimum cardinality $k$-adjacency generator is called a $k$-adjacency basis of $G$ and its cardinality, the $k$-adjacency dimension of $G$. In this article we study the problem of finding the $k$-adjacency dimension of a graph. We give some necessary and sufficient conditions for the existence of a $k$-adjacency basis of an arbitrary graph $G$ and we obtain general results on the $k$-adjacency dimension, including general bounds and closed formulae for some families of graphs. In particular, we obtain closed formulae for the $k$-adjacency dimension of join graphs $G+H$ in terms of the $k$-adjacency dimension of $G$ and $H$. These results concern the $k$-metric dimension, as join graphs have diameter two. As we can expect, the obtained results will become important tools for the study of the $k$-metric dimension of lexicographic product graphs and corona product graphs

The kk-metric dimension of corona product graphs

Given a connected simple graph $G=(V,E)$, and a positive integer $k$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if and only if for any pair of different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$, for every $i\in \{1,...,k\}$, where $d_G(x,y)$ is the length of a shortest path between $x$ and $y$. A $k$-metric generator of minimum cardinality in $G$ is called a $k$-metric basis and its cardinality, the $k$-metric dimension of $G$. In this article we study the $k$-metric dimension of corona product graphs $G\odot\mathcal{H}$, where $G$ is a graph of order $n$ and $\mathcal{H}$ is a family of $n$ non-trivial graphs. Specifically, we give some necessary and sufficient conditions for the existence of a $k$-metric basis in a connected corona graph. Moreover, we obtain tight bounds and closed formulae for the $k$-metric dimension of connected corona graphs.Comment: 22 page

The k-metric dimension of graphs: a general approach

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 d(v,w_i),\; \mbox{\rm for all}\; i\in \{1, \ldots k\}. Let $\mathcal{R}_k(X)$ be the set of metric generators for $X$. The $k$-metric dimension $\dim_k(X)$ of $(X,d)$ is defined as $$\dim_k(X)=\inf\{|S|:\, S\in \mathcal{R}_k(X)\}.$$ Here, we discuss the $k$-metric dimension of $(V,d_t)$, where $V$ is the set of vertices of a simple graph $G$ and the metric $d_t:V\times V\rightarrow \mathbb{N}\cup \{0\}$ is defined by $d_t(x,y)=\min\{d(x,y),t\}$ from the geodesic distance $d$ in $G$ and a positive integer $t$. The case $t\ge D(G)$, where $D(G)$ denotes the diameter of $G$, corresponds to the original theory of $k$-metric dimension and the case $t=2$ corresponds to the theory of $k$-adjacency dimension. Furthermore, this approach allows us to extend the theory of $k$-metric dimension to the general case of non-necessarily connected graphs

The kk-metric dimension of the lexicographic product of graphs

Given a simple and connected graph $G=(V,E)$, and a positive integer $k$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$, if for any pair of different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,\ldots,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$, for every $i\in \{1,\ldots,k\}$, where $d_G(x,y)$ denotes the distance between $x$ and $y$. The minimum cardinality of a $k$-metric generator is the $k$-metric dimension of $G$. A set $S\subseteq V$ is a $k$-adjacency generator for $G$ if any two different vertices $x,y\in V(G)$ satisfy $|((N_G(x)\triangledown N_G(y))\cup\{x,y\})\cap S|\ge k$, where $N_G(x)\triangledown N_G(y)$ is the symmetric difference of the neighborhoods of $x$ and $y$. The minimum cardinality of any $k$-adjacency generator is the $k$-adjacency dimension of $G$. In this article we obtain tight bounds and closed formulae for the $k$-metric dimension of the lexicographic product of graphs in terms of the $k$-adjacency dimension of the factor graphs.Comment: 19 page

The strong metric dimension of some generalized Petersen graphs

In this paper the strong metric dimension of generalized Petersen graphs $GP(n,2)$ is considered. The exact value is determined for cases $n=4k$ and $n=4k+2$, while for $n=4k+1$ an upper bound of the strong metric dimension is presented

The Simultaneous Metric Dimension of Families Composed by Lexicographic Product Graphs

Let ${\mathcal G}$ be a graph family defined on a common (labeled) vertex set $V$. A set $S\subseteq V$ is said to be a simultaneous metric generator for ${\cal G}$ if for every $G\in {\cal G}$ and every pair of different vertices $u,v\in V$ there exists $s\in S$ such that $d_{G}(s,u)\ne d_{G}(s,v)$, where $d_{G}$ denotes the geodesic distance. A simultaneous adjacency generator for ${\cal G}$ is a simultaneous metric generator under the metric $d_{G,2}(x,y)=\min\{d_{G}(x,y),2\}$. A minimum cardinality simultaneous metric (adjacency) generator for ${\cal G}$ is a simultaneous metric (adjacency) basis, and its cardinality the simultaneous metric (adjacency) dimension of ${\cal G}$. Based on the simultaneous adjacency dimension, we study the simultaneous metric dimension of families composed by lexicographic product graphs