Some planar graphs with ten-sided faces and their metric dimension

Abstract

Let ? = (V, E) be a non-trivial planar connected graph with vertex set V and edge set E. A set of ordered vertices R from V (?) is said to be a resolving set for ? if each vertex of ? is uniquely determined by its vector of distances to the vertices of R. The number of vertices in a smallest resolving set is called the metric dimension of ?. In this article, we study the metric dimension for two families of planar graphs, each of which is shown to have an independent minimum resolving set with cardinality three.