Let G = (V, E) be a connected graph, let v ∈ V be a vertex and let e = uw ∈ E be an edge. The distance between the vertex v and the edge e is given by. A set S of vertices in a connected graph G is an edge metric generator for G if every two edges of G are distinguished by some vertex of S. The smallest cardinality of an edge metric generator for G is called the edge metric dimension and is denoted by edim(G). In this article we introduce the concept of edge metric dimension and initiate the study of its mathematical properties. We make a comparison between the edge metric dimension and the standard metric dimension of graphs while presenting some realization results concerning the edge metric dimension and the standard metric dimension of graphs. We prove that computing the edge metric dimension of connected graphs is NP-hard and give some approximation results. Moreover, we present some bounds and closed formulae for the edge metric dimension of several classes of graphs.

The study and analysis of social graphs impacts on a wide range of applications, such as community decision making support and recommender systems. With the boom of online social networks, such analyses are benefiting from a massive collection and publication of social graphs at large scale. Unfortunately, individuals' privacy right might be inadvertently violated when publishing this type of data. In this article, we introduce (k, )-anonymity; a novel privacy measure aimed at evaluating the resistance of social graphs to active attacks. (k, )-anonymity is based on a new problem in Graph Theory, the k-metric antidimension defined as follows.Let G = (V, E) be a simple connected graph and S = {w 1 , · · · , w t } ⊆ V an ordered subset of vertices. The metric representation of a vertex u ∈ V with respect to S is the t-vector r(u|S) w t )), where d G (u, v) represents the length of a shortest u − v path in G. We call S a k-antiresolving set if k is the largest positive integer such that for every vertex v ∈ V − S there exist other k − 1 different vertices v 1 , · · · , v k−1 ∈ V − S with r(v|S) = r(v 1 |S) = · · · = r(v k−1 |S). The k-metric antidimension of G is the minimum cardinality among all the k-antiresolving sets for G.We address the k-metric antidimension problem by proposing a true-biased algorithm with success rate above 80% when considering random graphs of size at most 100. The proposed algorithm is used to determine the privacy guarantees offered by two real-life social graphs with respect to (k, )-anonymity. We also investigate theoretical properties of the k-metric antidimension of graphs. In particular, we focus on paths, cycles, complete bipartite graphs and trees.

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 $d(v,v_i)$, $i\in \{1,...,k\}$ denotes the distance between $v$ and $v_i$. $S$ is a resolving set for $G$ if for every pair of vertices $u,v$ of $G$, $r(u|S)\ne r(v|S)$. The metric dimension of $G$, $dim(G)$, is the minimum cardinality of any resolving set for $G$. Let $G$ and $H$ be two graphs of order $n_1$ and $n_2$, respectively. The corona product $G\odot H$ is defined as the graph obtained from $G$ and $H$ by taking one copy of $G$ and $n_1$ copies of $H$ and joining by an edge each vertex from the $i^{th}$-copy of $H$ with the $i^{th}$-vertex of $G$. For any integer $k\ge 2$, we define the graph $G\odot^k H$ recursively from $G\odot H$ as $G\odot^k H=(G\odot^{k-1} H)\odot H$. We give several results on the metric dimension of $G\odot^k H$. For instance, we show that given two connected graphs $G$ and $H$ of order $n_1\ge 2$ and $n_2\ge 2$, respectively, if the diameter of $H$ is at most two, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(H)$. Moreover, if $n_2\ge 7$ and the diameter of $H$ is greater than five or $H$ is a cycle graph, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(K_1\odot H).

Let G = (V, E) be a connected graph. A vertex w ∈ V distinguishes two elements (vertices or edges) x, y ∈ E ∪ V if d G (w, x) = d G (w, y). A set S of vertices in a connected graph G is a mixed metric generator for G if every two elements (vertices or edges) of G are distinguished by some vertex of S. The smallest cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by mdim(G). In this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.

Given a connected simple graph G = (V, E), and a positive integer k, a set S ⊆ V is said to be a k-metric generator for G if and only if for any pair of different vertices u, v ∈ V , there exist at least k vertices w 1 , w 2 , ...,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 ⊙ H, where G is a graph of order n and 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.

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 set for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $W$. The smallest cardinality of a strong resolving set for $G$ is called the strong metric dimension of $G$. It is known that the problem of computing this invariant is NP-hard. It is therefore desirable to reduce the problem of computing the strong metric dimension of product graphs, to the problem of computing some parameter of the factor graphs. We show that the problem of finding the strong metric dimension of the corona product $G\odot H$, of two graphs $G$ and $H$, can be transformed to the problem of finding certain clique number of $H$. As a consequence of the study we show that if $H$ has diameter two, then the strong metric dimension of $G\odot H$ is obtained from the strong metric dimension of $H$ and, if $H$ is not connected or its diameter is greater than two, then the strong metric dimension of $G\odot H$ is obtained from the strong metric dimension of $K_1\odot H$, where $K_1$ denotes the trivial graph. The strong metric dimension of join graphs is also studied

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 set for G if every pair of vertices of G is strongly resolved by some vertex of W . The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. It is known that the problem of computing the strong metric dimension of a graph is NP-hard. In this paper we obtain closed formulae for the strong metric dimension of several families of Cartesian product graphs and direct product 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 $d(u,w_i)\ne d(v,w_i),\; \textrm{for all}\; i\in \{1, \ldots k\}.$ 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 $$\begin{equation*}\dim_k(X)=\inf\{|S|:\, S\in \mathcal{R}_k(X)\}.\end{equation*}$$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. Finally, we analyse the computational complexity of determining the $k$-metric dimension of $(V,d_t)$ for the metric $d_t$.