# On Commutative Characterization of Graph Operation with Respect to Metric Dimension

## Authors

• Liliek Susilowati Department of Mathematics, Faculty of Sciences and Technology, Airlangga University, Jl. Mulyorejo, Surabaya 60115
• Mohammad Imam Utoyo Department of Mathematics, Faculty of Sciences and Technology, Airlangga University, Jl. Mulyorejo, Surabaya 60115
• Slamin Slamin Study Program of Information System, Jember University, Jl. Kalimantan 37, Jember 68121

## Keywords:

comb product, commutative with respect to metric dimension, corona product, generalized comb and corona products, metric dimension basis

## Abstract

Let G be a connected graph with vertex set V(G) and W={w1, w2, ..., wm} ?V(G). A representation of a vertex v? V(G) with respect to W is an ordered m-tuple r(v|W)=(d(v,w1),d(v,w2),...,d(v,wm)) where d(v,w) is the distance between verticesv and w. The set W is called a resolving set for G if every vertex of G has a distinct representation with respect to W. A resolving set containing a minimum number of vertices is called a basis for G. The metric dimension of G, denoted by dim (G), is the number of vertices in a basis of G. In general, the comb product and the corona product are non-commutative 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.

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