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

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.

[1] G.Chartrand, L. Eroh, M.A. Johnson, and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., 105 (200), 99–113.

[2] R.Frucht and F. Harary, On the coronaoftwographs, AequationesMathematicae, 4(3)(1970) , 322-325.

[3] C.D.Godsiland B.D. McKay, A new graph product and its spectrum, Bulletin of the Australian Mathematical Society, 18 (1978), 21-28.

[4] F.Okamoto, L. Crosse, B. Phinezy, P. Zhang, and Kalamazo, The local metric dimension of graph, Mathematica Bohemica, 135(3)( 2010): 239 – 255.

[5] J.A.Rodriguez-Velazquez, G.A. Barragan-Ramirez, and C.G. Gomez, On the local metric dimension of corona product graphs, Combinatorial And Computational Results, Arxiv: 1308.6689.v1 (Math Co) (2013), 30 Agustus.

[6] J.A.Rodriguez-Velazquez, C.G.Gomez., and G.A. Barragan-Ramirez, Computing the local metric dimension of graph from the local metric dimension of primary subgraph, arxiv:1402.0177v1[math. CO](2014), 2 Feb.

[7] S.W.Saputro, N.Mardiana, and I.A. Purwasi, The metric dimension of comb product graph, Graph Theory Conference in Honor of Egawa’s 60th Birthday, September 10 to 14, [internet] [citation 22 October 2013]: http://www.rs.tus.ac.jp/egawa_60th_birthday/abstract/contributed_talk/Suhadi_Wido_Saputro.

[8] L.Susilowati, Slamin, M.I. Utoyo, and N. Estuningsih, the similarity of metric dimension and local metric dimension of rooted product graph, Far East Journal of Mathematical Sciences, 97(7) (2015), 841-856.

[9] L.Susilowati, M.I. Utoyo, and Slamin, On commutative characterization of generalized comb and corona products of graphs with respect to thelocal metric dimension, Far East Journal of Mathematical Sciences, 100(4) (2016), 643-660.

[10] L.Susilowati, M.I. Utoyo, and Slamin, On commutative characterization of graph operation with respect to metric dimension, Journal of Mathematical and Fundamental Sciences, Vol. 49, No. 2, 2017, 156-170.

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