Proceedings of the SEMANTIK Conference of Mathematics Education (SEMANTIK 2019)

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

Authors
Citra Ayu Ratna Saidah, Tri Atmojo Kusmayadi
Corresponding Author
Citra Ayu Ratna Saidah
Available Online 28 August 2020.
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.

Open Access

Volume Title
Proceedings of the SEMANTIK Conference of Mathematics Education (SEMANTIK 2019)
Series
Advances in Social Science, Education and Humanities Research
Publication Date
28 August 2020
ISBN
10.2991/assehr.k.200827.110
ISSN
2352-5398
DOI
10.2991/assehr.k.200827.110How to use a DOI?
Open Access

TY  - CONF
AU  - Citra Ayu Ratna Saidah