The dominant metric dimension of graphs

Liliek Susilowati; Imroatus Sa'adah; Ratna Zaidatul Fauziyyah; Ahmad Erfanian; Slamin

doi:10.1016/j.heliyon.2020.e03633

The dominant metric dimension of graphs

Heliyon. 2020 Mar 23;6(3):e03633. doi: 10.1016/j.heliyon.2020.e03633. eCollection 2020 Mar.

Authors

Liliek Susilowati¹, Imroatus Sa'adah¹, Ratna Zaidatul Fauziyyah¹, Ahmad Erfanian², Slamin³

Affiliations

¹ Mathematics Department, Faculty of Science and Technology, Airlangga University, Indonesia.
² Department of Pure Mathematics, Ferdowsi University of Mashhad, Iran.
³ Study Program of Informatics, Faculty of Computer Science, Universitas Jember, Indonesia.

Abstract

The G be a connected graph with vertex set $V (G)$ and edge set $E (G)$ . A subset $S \subseteq V (G)$ is called a dominating set of G if for every vertex x in $V (G) ∖ S$ , there exists at least one vertex u in S such that x is adjacent to u. An ordered set $W \subseteq V (G)$ is called a resolving set of G, if every pair of vertices u and v in $V (G)$ have distinct representation with respect to W. An ordered set $S \subseteq V (G)$ is called a dominant resolving set of G, if S is a resolving set and also a dominating set of G. The minimum cardinality of dominant resolving set is called a dominant metric dimension of G, denoted by $D d i m (G)$ . In this paper, we investigate the dominant metric dimension of some particular class of graphs, the characterisation of graph with certain dominant metric dimension, and the dominant metric dimension of joint and comb products of graphs.

Keywords: Dominant metric dimension; Dominant resolving set; Dominating set; Mathematics; Metric dimension; Resolving set.