## The dominant metric dimension of graphs

Published in Heliyon

Mathematics; Metric dimension; Resolving set; Dominating set; Dominant resolving set; Dominant metric dimension

Published in Heliyon

Mathematics; Metric dimension; Resolving set; Dominating set; Dominant resolving set; Dominant metric dimension

In the localization game, introduced by Seager in 2013, an invisible and immobile target is hidden at some vertex of a graph $G$. At every step, one vertex $v$ of $G$ can be probed which results in the knowledge of the distance between $v$ and the secret location of the target. The objective of the game is to minimize the number of steps needed to ...

Published in Journal of mathematical biology

Numerous data analysis and data mining techniques require that data be embedded in a Euclidean space. When faced with symbolic datasets, particularly biological sequence data produced by high-throughput sequencing assays, conventional embedding approaches like binary and k-mer count vectors may be too high dimensional or coarse-grained to learn fro...

The metric dimension $MD(G)$ of an undirected graph $G$ is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of $G$. Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs. In this work, we study...

Concept of resolving set and metric basis has enjoyed a lot of success because of multipurpose applications both in computer and mathematical sciences. A system in which failure of any single unit, another chain of units not containing the faulty unit can replace the originally used chain is called fault-tolerant self-stable system. Recent research...

The metric dimension $MD(G)$ of an undirected graph $G$ is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of $G$. Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs. In this work, we study...

Let G = (V, E) be a connected graph and d(u, v) denote the distance between the vertices u and v in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). Let J2n,m be a m-le...

Published in Journal of Combinatorial Optimization

For an undirected graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E)$$\end{document}, a vertex τ∈V\documentclass[12pt]{minimal} \usepackage{amsmath} \...

Published in Algorithmica

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Open Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the...

We study the problems Locating-Dominating Set and Metric Dimension, which consist of determining a minimum-size set of vertices that distinguishes the vertices of a graph using either neighbourhoods or distances. We consider these problems when restricted to interval graphs and permutation graphs. We prove that both decision problems are NP-complet...