On Metric Dimension of Some Rotationally Symmetric Graphs

A family of connected graphs is a family with constant metric dimension if dim( ) is finite and does not depend upon the choice of in . In this paper, we show that the graph and the graph obtained from the antiprism graph have constant metric dimension. 1. Notation and Preliminary Results For a connected graph , the distance between two vertices is the length of a shortest path between them. A vertex of a graph is said to resolve two vertices, and , of if . Let be an ordered set of vertices of and let be a vertex of . The representation of the vertex with respect to denoted by is the -tuple . If distinct vertices of have distinct representations with respect to , then is called a resolving set, for [1]. A resolving set of minimum cardinality is called a metric basis for , and the cardinality of this set is the metric dimension of , denoted by . For a given ordered set of vertices of a graph , the th component of is if and only if . Thus, to show that is a resolving set it suffices to verify that for each pair of distinct vertices . Caceres et al. [2] found the metric dimension of fan , and Javaid et al. [3] found the metric dimension of Jahangir graph . In [1], Chartrand et al. proved that a graph has metric dimension if and only if it is a path; hence paths on vertices constitute a family of graphs with constant metric dimension. They also showed that cycles with vertices also constitute such a family of graphs as their metric dimension is . In [2], Caceres et al. proved that Prisms are the trivalent plane graphs obtained by the cartesian product of the path with a cycle . In [4], Javaid et al. proved that this family has constant metric dimension. In [4], Javaid et al. also proved that the antiprism graph , constitutes a family of regular graphs with constant metric dimension as , for every . In this paper, we extend this study to antiprism-related graphs (see Figure 1). The graph is defined as follows: for each vertex , of the outer cycle of the antiprism graph, we introduce a new vertex , and join to and , , with taken modulo . Thus, . Here ？？ are the inner cycle vertices, ？？ are the outer cycle vertices, and ？？ are adjacent vertices to the outer cycle. We define the graph as follows: for each vertex , of the outer cycle of the antiprism graph, we introduce a new vertex and join to , . Thus, . Here, are the inner cycle vertices, are the outer cycle vertices, are the pendant vertices adjacent to the outer cycle vertices. Figure 1: Graphs and . 2. Antiprism-Related Graphs with Constant Metric Dimension In this section, we show that , have constant