The Generalization of Graph Operations and Their Local Metric Dimension

Liliek Susilowati; Mohammad Imam Utoyo; Slamin

doi:https://doi.org/10.14445/22315373/IJMTT-V65I4P505

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 65 | Issue 4 | Year 2019 | Article Id. IJMTT-V65I4P505 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I4P505

The Generalization of Graph Operations and Their Local Metric Dimension

Liliek Susilowati, Mohammad Imam Utoyo, Slamin

Citation :

Liliek Susilowati, Mohammad Imam Utoyo, Slamin, "The Generalization of Graph Operations and Their Local Metric Dimension," International Journal of Mathematics Trends and Technology (IJMTT), vol. 65, no. 4, pp. 22-27, 2019. Crossref, https://doi.org/10.14445/22315373/IJMTT-V65I4P505

Abstract

Let 𝐺 be a connected graph with vertex set V(G) and 𝑊 = {𝑤1, 𝑤2, … , 𝑤𝑚 } ⊂ 𝑉(𝐺). The representation of a vertex 𝑣 ∈ 𝑉(𝐺) with respect to 𝑊 is the ordered m-tuple 𝑟(𝑣|𝑊) = (𝑑 𝑣, 𝑤1 , 𝑑 𝑣, 𝑤2, . . . , 𝑑 𝑣, 𝑤𝑚 ) where 𝑑(𝑣, 𝑤) represents the distance between vertices 𝑣 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 basis for 𝐺. The metric dimension of 𝐺, denoted by𝑑𝑖𝑚(𝐺), is the number of vertices in a basis of 𝐺.The set 𝑊 is called a local resolving set for 𝐺 if every two adjacent vertices of 𝐺 have a distinct representation and a minimum local resolving set is called a local basis of G. The cardinality of a local basis of G is called the local metric dimension of G, denoted by di ml(G). The comb product and the corona product are noncommutative operations in graph, but these operations can be commutative with respect to the local metric dimension for some graphs with certain conditions. In this paper, we determine the local metric dimension of the most generalized comb and corona products of graphs. Futher more, we determine the commutative characterization of comb and corona products with respect to the local metric dimension.

Keywords

resolving set, basis, local basis, local metric dimension,the most generalizedcomb and corona products of graph,commutative characterization.

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