### DIMENSI METRIK KETETANGGAAN LOKAL GRAF HASIL OPERASI k-COMB

#### Abstract

Research on the local adjacency metric dimension has not been found in all operations of the graph, one of them is comb product graph. The purpose of this research was to determine the local adjacency metric dimension of k-comb product graph and level comb product graph between any connected graph G and H. In this research graph G and graph H such as cycle graph, complete graph, path graph, and star graph. K-comb product graph between any graph G and H denoted by Go_{k}H. While level k comb product graph between any graph G and H denoted by Go^{k}H.

In this research, local adjacency metric dimension of Go_{k}S_{m} graph only dependent to multiplication of the cardinality of V(G) and many of k value, while Go_{k}K_{m} graph and Go_{k}C_{m} graph is dependent to dominating number of G and multiplication of the cardinality of V(G), many of k value, and local adjacency metric dimension of K_{m} graph or C_{m} graph. And then, local adjacency metric dimension of Go_{k}S_{m} graph only dependent to the cardinality of V(Go^{k-1}S_{m}), while Go^{k}K_{m} graph and Go^{k}C_{m} graph is dependent to dominating number of G and multiplication of the local adjacency metric dimension of K_{m} graph or C_{m} graph with cardinality of V(Go^{k-1}K_{m}) or V(Go^{k-1}C_{m}).

#### Keywords

#### References

Iswadi, H., Baskoro, E.T., Simanjutak, R., 2011, On The Metric Dimension of Corona Product Graphs, Far East Journal of Mathematical Sciences (FJMS), 52 (2): 155-170.

Okamoto, F., Crosse, L., Phinezy, B., Zhang, P., dan Kalamazo, 2010, The Local Metrik Dimension of Graph, Mathematica Bohemica, 135(3): 239-255.

Jannesari, M. dan Omoomi, B., 2012, The Metrik Dimension of The Lexicographic Product of Graphs, Discrete Mathematics, Vol. 312, No. 22, pp. 3349-3356.

Rodriguez-Velazquez, Gomez. C.G., dan Barragan –Ramirez, G. A., 2014, Computing the Local Metric Dimension of Graph from The Local Metric Dimension of Primary Subgraph, arvix: 1402.0177v1[math.CO], 2 Februari 2014.

Rodriguez-Velazquez, J. A. dan Fernau, H., 2016, On the (Adjacency) Metrik Dimension of Corona and Strong Product Graphs and Their Local Variants: Combinatorial and Computational Results.

Susilowati, L., Salamin, Utoyo, M. I. dan Estuningsih, N., 2016, The Similarity of Metric Dimension and Local Metric Dimension of Rooted Product Graph, Far East Journal of Mathematical Sciences (FJMS), Volume 97, Number 7, Pages 841-856.

Saputro, S. W., Mardina, N., dan Purwasi, LA., 2013, The Metrik Dimension of Comb Product Graph, Graph Theory Conferience in Honor of Egawa’s 60th Birthday.

Susilowati, L. dan Utoyo, M. I., 2016, On Comutative Characterization of Generalized Comb and Corona Products of Graph with Respect to the Local Metric Dimension, Far East Journal of Mathematical Sciences (FJMS), Volume 100.

Munawwaroh, Ana Zahrotul, 2016, Dimensi Metrik Fraksional Generalisasi Hasil Kali Comb dari Graf Lengkap, Repository Unair.

### Refbacks

- There are currently no refbacks.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.