A note on the metric dimension of subdivided thorn graphs

Lyra Yulianti, Narwen Narwen, Sri Hariyani

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(vW) = {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(uW) ≠ r(vW). 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 ≤ in. 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.


Keywords


metric dimension; thorn-subdivided graphs; complete graph

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DOI: http://dx.doi.org/10.19184/ijc.2019.3.1.4

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