The k-Metric Dimension of Nk + Pn Graph and Starbarbell Graph

DOI

10.2991/assehr.k.200827.110How to use a DOI?

Keywords

k-metric dimension, k-metric generator, basis of k-metric, Nk + Pn graph, starbarbell graph

Abstract

Let G be a simple connected graph with a set of vertices V(G) and set of edges E(G). The distance between two vertices u and v in a graph G are the shortest path length between two vertices u and v denoted by d(u,v). Let k be a positive integer, S ⊆ V with S is a k-metric generator if and only if for each different vertex pair u,v ∈ V there are at least k vertices w1,w2, …, wk ∈ S and fulfill d(u,wi) ≠ d(v,wi) with i ∈ {1,2,…,k}. Minimum cardinality of a k-metric generator of a graph G is called the basis k-metric of graph G. The number of elements on the basis of k-metric graph G are called k-metric dimension of graph G and denoted by dimk(G). Nk + Pn is the result of a join operation between null graph Nk and path graph Pn with k,n ≥ 2. Starbarbell graph denoted by SBm1,m2,…,mn is a graph formed from a star graph K1,n and n complete graph Kmi then merge one vertex from each Kmi with ith leaf of K1,n with mi ≥ 3, 1 ≤ i ≤ n, and n ≥ 2. In this paper, we determine the k-metric dimension of Nk + Pn graph and starbarbell graph.

Copyright

© 2020, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - CONF
AU  - Citra Ayu Ratna Saidah
AU  - Tri Atmojo Kusmayadi
PY  - 2020
DA  - 2020/08/28
TI  - The k-Metric Dimension of Nk + Pn Graph and Starbarbell Graph
BT  - Proceedings of the SEMANTIK Conference of Mathematics Education (SEMANTIK 2019)
PB  - Atlantis Press
SP  - 18
EP  - 22
SN  - 2352-5398
UR  - https://doi.org/10.2991/assehr.k.200827.110
DO  - 10.2991/assehr.k.200827.110
ID  - Saidah2020
ER  -