arXiv subjects

Ismael G. Yero

Publications and source records attributed to Ismael G. Yero.

At least 73 records · Page 4Linked to original sources

arXiv2018

Packing and domination parameters in digraphs

Given a digraph $D=(V,A)$, a set $B\subset V$ is a packing set in $D$ if there are no arcs joining vertices of $B$ and for any two vertices $x,y\in B$ the sets of in-neighbors of $x$ and $y$ are disjoint. The set $S$ is a dominating set (an open dominating set) in $D$ if every vertex not in $S$ (in $V$) has an in-neighbor in $S$. Moreover, a dominating set $S$ is called a total dominating set if the subgraph induced by $S$ has no isolated vertices. The packing sets of maximum cardinality and the (total, open) dominating sets of minimum cardinality in digraphs are studied in this article. We prove that the two optimal sets concerning packing and domination achieve the same value for directed trees, and give some applications of it. We also show analogous equalities for all connected contrafunctional digraphs, and characterize all such digraphs $D$ for which such equalities are satisfied. Moreover, sharp bounds on the maximum and the minimum cardinalities of packing and dominating sets, respectively, are given for digraphs. Finally, we present solutions for two open problems, concerning total and open dominating sets of minimum cardinality, pointed out in [Australas. J. Combin. 39 (2007), 283--292].

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arXiv2017

On computational and combinatorial properties of the total co-independent domination number of graphs

A subset $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total dominating set $D$ is called a total co-independent dominating set if the subgraph induced by $V-D$ is edgeless and has at least one vertex. The minimum cardinality of any total co-independent dominating set is the total co-independent domination number of $G$ and is denoted by $γ_{t,coi}(G)$. In this work we study some complexity and combinatorial properties of $γ_{t,coi}(G)$. Specifically, we prove that deciding whether $γ_{t,coi}(G)\le k$ for a given integer $k$ is an NP-complete problem and give several bounds on $γ_{t,coi}(G)$. Also, since any total co-independent dominating set is also a total dominating set, we characterize all the trees having equal total co-independent domination number and total domination number.

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arXiv2017

Error-correcting codes from $k$-resolving sets

We demonstrate a construction of error-correcting codes from graphs by means of $k$-resolving sets, and present a decoding algorithm which makes use of covering designs. Along the way, we determine the $k$-metric dimension of grid graphs (i.e. Cartesian products of paths).

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arXiv2017

Open k-monopolies in graphs: complexity and related concepts

Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open $k$-monopolies in graphs which are closely related to different parameters in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $δ_X(v)$ is the number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be $k$-controlled by $M$ if $δ_M(v)\ge \frac{δ_V(v)}{2}+k$. The set $M$ is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of $G$. - A function $f: V\rightarrow \{-1,1\}$ is called a signed total $t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all $v\in V$. - A nonempty set $S\subseteq V$ is a global (defensive and offensive) $k$-alliance in $G$ if $δ_S(v)\ge δ_{V-S}(v)+k$ holds for every $v\in V$. In this article we prove that the problem of computing the minimum cardinality of an open $0$-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open $k$-monopolies and we derive some exact values.

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arXiv2016

Mixed metric dimension of graphs

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 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 $\mathrm{mdim}(G)$. 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.

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arXiv2016

The fractional strong metric dimension in three graph products

For any two distinct vertices $x$ and $y$ of a graph $G$, let $S\{x, y\}$ denote the set of vertices $z$ such that either $x$ lies on a $y-z$ geodesic or $y$ lies on an $x-z$ geodesic. Let $g: V(G) \rightarrow [0,1]$ be a real valued function and, for any $U \subseteq V(G)$, let $g(U)=\sum_{v \in U}g(v)$. The function $g$ is a strong resolving function of $G$ if $g(S\{x, y\}) \ge 1$ for every pair of distinct vertices $x, y$ of $G$. The fractional strong metric dimension, $sdim_f(G)$, of a graph $G$ is $\min\{g(V(G)): g \mbox{ is a strong resolving function of }G\}$. In this paper, after obtaining some new results for all connected graphs, we focus on the study of the fractional strong metric dimension of the corona product, the lexicographic product, and the Cartesian product of graphs.

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arXiv2016

On the Computational Complexities of Three Privacy Measures for Large Networks Under Active Attack

With the arrival of modern internet era, large public networks of various types have come to existence to benefit the society as a whole and several research areas such as sociology, economics and geography in particular. However, the societal and research benefits of these networks have also given rise to potentially significant privacy issues in the sense that malicious entities may violate the privacy of the users of such a network by analyzing the network and deliberately using such privacy violations for deleterious purposes. Such considerations have given rise to a new active research area that deals with the quantification of privacy of users in large networks and the corresponding investigation of computational complexity issues of computing such quantified privacy measures. In this paper, we formalize three such privacy measures for large networks and provide non-trivial theoretical computational complexity results for computing these measures. Our results show the first two measures can be computed efficiently, whereas the third measure is provably hard to compute within a logarithmic approximation factor. Furthermore, we also provide computational complexity results for the case when the privacy requirement of the network is severely restricted, including an efficient logarithmic approximation.

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arXiv2016

Uniquely identifying the edges of a graph: the edge metric dimension

Let $G=(V,E)$ be a connected graph, let $v\in V$ be a vertex and let $e=uw\in E$ be an edge. The distance between the vertex $v$ and the edge $e$ is given by $d_G(e,v)=\min\{d_G(u,v),d_G(w,v)\}$. A vertex $w\in V$ distinguishes two edges $e_1,e_2\in E$ if $d_G(w,e_1)\ne d_G(w,e_2)$. 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.

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arXiv2015

Graphs that are simultaneously efficient open domination and efficient closed domination graphs

A graph is an efficient open (resp.\ closed) domination graph if there exists a subset of vertices whose open (resp.\ closed) neighborhoods partition its vertex set. Graphs that are efficient open as well as efficient closed (shortly EOCD graphs) are investigated. The structure of EOCD graphs with respect to their efficient open and efficient closed dominating sets is explained. It is shown that the decision problem regarding whether a graph is an EOCD graph is an NP-complete problem. A recursive description that constructs all EOCD trees is given and EOCD graphs are characterized among the Sierpiński graphs.

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arXiv2015

On the complexity of computing the $k$-metric dimension of graphs

Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is a $k$-metric generator for $G$ if for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. We study some problems regarding the complexity of some $k$-metric dimension problems. For instance, we show that the problem of computing the $k$-metric dimension of graphs is $NP$-Complete. However, the problem is solved in linear time for the particular case of trees.

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arXiv2015

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.

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arXiv2015

Resolvability and Strong Resolvability in the Direct Product of Graphs

Given a connected graph $G$, a vertex $w\in V(G)$ distinguishes two different vertices $u,v$ of $G$ if the distances between $w$ and $u$ and between $w$ and $v$ are different. Moreover, $w$ strongly resolves the pair $u,v$ 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) metric generator for $G$ if every pair of vertices of $G$ is (strongly resolved) distinguished by some vertex of $W$. The smallest cardinality of a (strong) metric generator for $G$ is called the (strong) metric dimension of $G$. In this article we study the (strong) metric dimension of some families of direct product graphs.

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arXiv2015

k-Metric Antidimension: a Privacy Measure for Social Graphs

Let $G = (V, E)$ be a simple connected graph and $S = \{w_1, \cdots, w_t\} \subseteq V$ an ordered subset of vertices. The metric representation of a vertex $u\in V$ with respect to $S$ is the $t$-vector $r(u|S) = (d_G(u, w_1), \cdots, d_G(u, w_t))$, where $d_G(u, v)$ represents the length of a shortest $u-v$ path in $G$. The set $S$ is called a resolving set for $G$ if $r(u|S) = r(v|S)$ implies $u = v$ for every $u, v \in V$. The smallest cardinality of a resolving set is the metric dimension of $G$. In this article we propose, to the best of our knowledge, a new problem in Graph Theory that resembles to the aforementioned metric dimension problem. We call $S$ a $k$-antiresolving set if $k$ is the largest positive integer such that for every vertex $v \in V-S$ there exist other $k-1$ different vertices $v_1, \cdots, v_{k-1} \in V-S$ with $r(v|S) = r(v_1|S) = \cdots = r(v_{k-1}|S)$, \emph{i.e.}, $v$ and $v_1, \cdots, v_{k-1}$ have the same metric representation with respect to $S$. The $k$-metric antidimension of $G$ is the minimum cardinality among all the $k$-antiresolving sets for $G$. In this article, we introduce a novel privacy measure, named $(k, \ell)$-anonymity and based on the $k$-metric antidimension problem, aimed at evaluating the resistance of social graphs to active attacks. We, therefore, propose a true-biased algorithm for computing the $k$-metric antidimension of random graphs. The success rate of our algorithm, according to empirical results, is above $80 \%$ and $90 \%$ when looking for a $k$-antiresolving basis and a $k$-antiresolving set respectively. We also investigate theoretical properties of the $k$-antiresolving sets and the $k$-metric antidimension of graphs. In particular, we focus on paths, cycles, complete bipartite graphs and trees.

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arXiv2015

The k-metric dimension of a graph

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 any pair of different vertices of $G$ are distinguished by at least $k$ elements of $S$, i.e., for any two 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 $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. A connected graph $G$ is $k$-metric dimensional if $k$ is the largest integer such that there exists a $k$-metric basis for $G$. We give a necessary and sufficient condition for a graph to be $k$-metric dimensional and we obtain several results on the $k$-metric dimension.

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arXiv2014

Closed formulae for the strong metric dimension of lexicographic product graphs

Given a connected graph $G$, a vertex $w\in V(G)$ 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 relationships between the strong metric dimension of the lexicographic product of graphs and the strong metric dimension of its factor graphs.

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arXiv2014

Bondage number of grid graphs

The bondage number $b(G)$ of a nonempty graph $G$ is the cardinality of a smallest set of edges whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. Here we study the bondage number of some grid-like graphs. In this sense, we obtain some bounds or exact values of the bondage number of some strong product and direct product of two paths.

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arXiv2013

Strong metric dimension of rooted product 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. This suggests finding the strong metric dimension for special classes of graphs or obtaining good bounds on this invariant. In this paper we study the problem of finding exact values or sharp bounds for the strong metric dimension of rooted product of graphs and express these in terms of invariants of the factor graphs.

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arXiv2013

On the strong metric dimension of Cartesian and direct products of graphs

Let $G$ be a connected graph. A vertex $w$ {\em 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 {\em 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 {\em 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.

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