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

#### Abstract

*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

*V*

_{1}and

*V*

_{2}where an induced subgraph of G by

*V*

_{1}and

*V*

_{2}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

#### Full Text:

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

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