A note on the metric dimension of subdivided thorn graphs

For some ordered subset W = { w 1 , w 2 , ⋯, w t } 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 , w 1 ), d ( v , w 2 ), ⋯, d ( v , w t )} . 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 T h ( G , l 1 , l 2 , ⋯, l n ) , is constructed from G by adding l i leaves to vertex v i of G , for l i ≥ 1 and 1 ≤ i ≤ n . The subdivided-thorn graph, denoted by T D ( G , l 1 ( y 1 ), l 2 ( y 2 ), ⋯, l n ( y n )) , is constructed by subdividing every l i leaves of the thorn graph of G into a path on y i vertices. In this paper the metric dimension of thorn of complete graph, dim( T h ( K n , l 1 , l 2 , ⋯, l n )) , l i ≥ 1 are determined, partially answering the problem proposed by Iswadi et al . This paper also gives some conjectures for the lower bound of dim( T h ( G , l 1 , l 2 , ⋯, l n )) , for arbitrary connected graph G . Next, the metric dimension of subdivided-thorn of complete graph, dim( T D ( K n , l 1 ( y 1 ), l 2 ( y 2 ), ⋯, l n ( y n )) are determined and some conjectures for the lower bound of dim( T h ( G , l 1 ( y 1 ), l 2 ( y 2 ), ⋯, l n ( y n )) for arbitrary connected graph G are given.