On Commutative Characterization of Graph Operation with Respect to Metric Dimension

Abstract

, is the number of vertices in a basis of . In general, the comb product and the corona product are noncommutative operations in a graph. However, these operations can be commutative with respect to the metric dimension for some graphs with certain conditions. In this paper, we determine the metric dimension of the generalized comb and corona products of graphs and the necessary and sufficient conditions of the graphs in order for the comb and corona products to be commutative operations with respect to the metric dimension.

The Dominant Metric Dimension of Corona Product Graphs

“…The metric dimension of and denoted by ( ), is defined by ( ) = min{| |, ℎ } (1). The study of metric dimension concepts and its applications have been done by Careres (2), Yero (3), Iswadi (4), Saputro (5), and Susilowati (6) for cartesian product graphs, corona product graphs, path graph and comb product graph. The concept of dominating set was studied by Gupta (7), Reni Umilasari and Darmaji (8).…”

Fractional Local Metric Dimension of Comb Product Graphs

“…In (4) introduced the local metric0dimension of0graph, they defined the local0resolving set and the local0metric dimension of a0graph. In (5,6) studied the commutative0characterization0of graph operations with0respect to the local metric0dimension and metric dimension, respectively. The development of the metric0dimension is the fractional0metric0dimension.…”

The dominant metric dimension of graphs

“…Metric dimension of some operations of graphs have been obtained, namely, metric dimension of comb product graphs [7], joint product graphs [8], and corona product graphs [9]. Several operation proper-ties in graphs related to metric dimensions are also studied by [10], [11], and [12].…”