New results on metric-locating-dominating sets of graphs

A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distanc es from the elements of S , and the minimum cardinality of such a set is called the metri c-location- domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominatin g sets to other special sets: resolving sets, dominating sets, locating-dominating set s and doubly resolving sets. We first characterize classes of trees according to cer tain relationships between their metric-location-domination number and thei r metric dimension and domination number. Then, we show different methods to tran sform metric- locating-dominating sets into locating-dominating sets a nd doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them involving parameters that have not been related so farPostprint (published version

New results on metric-locating-dominating sets of graphs

A dominating set $S$ of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of $S$, and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize classes of trees according to certain relationships between their metric-location-domination number and their metric dimension and domination number. Then, we show different methods to transform metric-locating-dominating sets into locating-dominating sets and doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them involving parameters that have not been related so far.Comment: 13 pages, 3 figure

The equidistant dimension of graphs

A subset S of vertices of a connected graph G is a distance-equalizer set if for every two distinct vertices x,y¿V(G)\S there is a vertex w¿S such that the distances from x and y to w are the same. The equidistant dimension of G is the minimum cardinality of a distance-equalizer set of G. This paper is devoted to introduce this parameter and explore its properties and applications to other mathematical problems, not necessarily in the context of graph theory. Concretely, we first establish some bounds concerning the order, the maximum degree, the clique number, and the independence number, and characterize all graphs attaining some extremal values. We then study the equidistant dimension of several families of graphs (complete and complete multipartite graphs, bistars, paths, cycles, and Johnson graphs), proving that, in the case of paths and cycles, this parameter is related to 3-AP-free sets. Subsequently, we show the usefulness of distance-equalizer sets for constructing doubly resolving sets.Peer ReviewedPostprint (published version

Bounds for the Generalized Distance Eigenvalues of a Graph

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D(G) be the distance matrix, DL(G) be the distance Laplacian, DQ(G) be the distance signless Laplacian, and Tr(G) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by Dα(G) the generalized distance matrix, i.e., Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of S , and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize the extremal trees of the bounds that naturally involve metric-location-domination number, metric dimension and domination number. Then, we prove that there is no polynomial upper bound on the location-domination number in terms of the metric-location-domination number, thus extending a result of Henning and Oellermann. Finally, we show different methods to transform metric-locating-dominating sets into locating-dominating sets and doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them concerning parameters that have not been related so farPeer ReviewedPostprint (author's final draft

Steiner Distance in Product Networks

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In this paper, we investigate the Steiner distance and Steiner $k$-diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner $k$-diameter of some networks.Comment: 29 pages, 4 figure