The local metric dimension of split and unicyclic graphs

• Published in: Indonesian journal of combinatorics Vol. 6; no. 1; pp. 50 - 57
• Main Authors: Fitriani, Dinny, Rarasati, Anisa, Saputro, Suhadi Wido, Baskoro, Edy Tri
• Format: Journal Article
• Language: English
• Published: InaCombS; Universitas Jember; dan Universitas Indonesia 28.06.2022
• Subjects: local basis, local metric dimension, local resolving set, split graph, unicyclic graph
• ISSN: 2541-2205; 2541-2205
• DOI: 10.19184/ijc.2022.6.1.3

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.