### The local metric dimension of split and unicyclic graphs

#### Abstract

A set W is called a local resolving set of G if the distance of u and v to some elements of W are distinct for every two adjacent vertices u and v in G.  The local metric dimension of G is the minimum cardinality of a local resolving set of G.  A connected graph G is called a split graph if V(G) can be partitioned into two subsets V1 and V2 where an induced subgraph of G by V1 and V2 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.

#### Keywords

local basis, local metric dimension, local resolving set, split graph, unicyclic graph

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DOI: http://dx.doi.org/10.19184/ijc.2022.6.1.3

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