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All Journal Indonesian Journal of Combinatorics
Sri Hariyani
Andalas University
Computer Science & IT Decision Sciences, Operations Research & Management

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Journal : Indonesian Journal of Combinatorics

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On the subdivided thorn graph and its metric dimension Lyra Yulianti; Narwen Narwen; Sri Hariyani
Indonesian Journal of Combinatorics Vol 3, No 1 (2019)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Download Original | Original Source | Check in Google Scholar | Full PDF (236.79 KB) | DOI: 10.19184/ijc.2019.3.1.4

Abstract

For some ordered subset W = {w1, w2, ⋯, wt} of vertices in connected graph G, and for some vertex v in G, the metric representation of v with respect to W is defined as the t-vector r(v∣W) = {d(v, w1), d(v, w2), ⋯, d(v, wt)}. The set W is the resolving set of G if for every two vertices u, v in G, r(u∣W) ≠ r(v∣W). The metric dimension of G, denoted by dim(G), is defined as the minimum cardinality of W. Let G be a connected graph on n vertices. The thorn graph of G, denoted by Th(G, l1, l2, ⋯, ln), is constructed from G by adding li leaves to vertex vi of G, for li ≥ 1 and 1 ≤ i ≤ n. The subdivided-thorn graph, denoted by TD(G, l1(y1), l2(y2), ⋯, ln(yn)), is constructed by subdividing every li leaves of the thorn graph of G into a path on yi vertices. In this paper the metric dimension of thorn of complete graph, dim(Th(Kn, l1, l2, ⋯, ln)), li ≥ 1 are determined, partially answering the problem proposed by Iswadi et al . This paper also gives some conjectures for the lower bound of dim(Th(G, l1, l2, ⋯, ln)), for arbitrary connected graph G. Next, the metric dimension of subdivided-thorn of complete graph, dim(TD(Kn, l1(y1), l2(y2), ⋯, ln(yn)) are determined and some conjectures for the lower bound of dim(Th(G, l1(y1), l2(y2), ⋯, ln(yn)) for arbitrary connected graph G are given.
Co-Authors Narwen Narwen Yulianti, Lyra