The local metric dimension of split and unicyclic graphs

Abstract

A set <em>W</em> is called a local resolving set of <em>G</em> if the distance of <em>u</em> and <em>v</em> to some elements of <em>W</em> are distinct for every two adjacent vertices <em>u</em> and <em>v</em> in <em>G</em>. The local metric dimension of <em>G</em> is the minimum cardinality of a local resolving set of <em>G</em>. A connected graph <em>G</em> is called a split graph if <em>V</em>(<em>G</em>) can be partitioned into two subsets <em>V</em><sub>1</sub> and <em>V</em><sub>2</sub> where an induced subgraph of G by <em>V</em><sub>1</sub> and <em>V</em><sub>2</sub> is a complete graph and an independent set, respectively. We also consider a graph, namely the unicyclic graph which is a connected graph containing exactly one cycle. In this paper, we provide a general sharp bounds of local metric dimension of split graph. We also determine an exact value of local metric dimension of any unicyclic graphs.