Bounds for metric dimensions of generalized neighborhood corona graphs

Rinurwati; S E Setiawan; Slamin

doi:10.1016/j.heliyon.2021.e07433

Bounds for metric dimensions of generalized neighborhood corona graphs

Heliyon. 2021 Jun 28;7(7):e07433. doi: 10.1016/j.heliyon.2021.e07433. eCollection 2021 Jul.

Authors

Rinurwati¹, S E Setiawan¹, Slamin²

Affiliations

¹ Department of Mathematics, Institut Teknologi Sepuluh Nopember, Indonesia.
² Study Program of Informatics, Faculty of Computer Science, Universitas Jember, Indonesia.

Abstract

In this paper, the authors analysed metric dimensions of arbitrary graphs $G \tilde{★} \land_{i = 1}^{| V (G) |} H_{i}$ in which graphs $G, H_{1}, H_{2}, \dots, H_{| V (G) |}$ are non-trivial, $G$ is connected, and $\tilde{★}$ denotes generalized neighborhood corona operation. We found lower bounds of $\dim (G \tilde{★} \land_{i = 1}^{| V (G) |} H_{i})$ as function of $\dim_{A} (H_{i})$ where $\dim_{A} (H_{i})$ denotes adjacency metric dimensions of $H_{i}$ . We also found upper bounds of $\dim (G \tilde{★} \land_{i = 1}^{| V (G) |} H_{i})$ when G does not contain pair of false twin vertices. Furthermore, we found a characteristic of $\dim (G \tilde{★} \land_{i = 1}^{| V (G) |} H_{i})$ which indicates that our lower bounds are strict.

Keywords: Metric dimension; Neighborhood corona graph; Resolving set.