Identifying the Exact Value of the Metric Dimension and Edge Dimension of Unicyclic Graphs

Enqiang Zhu, Shaoxiang Peng and Chanjuan Liu ()
Additional contact information
Enqiang Zhu: Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
Shaoxiang Peng: Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
Chanjuan Liu: School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China

Mathematics, 2022, vol. 10, issue 19, 1-14

Abstract: Given a simple connected graph G , the metric dimension dim ( G ) (and edge metric dimension edim ( G ) ) is defined as the cardinality of a smallest vertex subset S ⊆ V ( G ) for which every two distinct vertices (and edges) in G have distinct distances to a vertex of S . It is an interesting topic to discuss the relation between these two dimensions for some class of graphs. This paper settles two open problems on this topic for unicyclic graphs. We recently learned that Sedlar and Škrekovski settled these problems, but our work presents the results in a completely different way. By introducing four classes of subgraphs, we characterize the structure of a unicyclic graph G such that dim ( G ) and edim ( G ) are equal to the cardinality of any minimum branch-resolving set for unicyclic graphs. This generates an approach to determine the exact value of the metric dimension (and edge metric dimension) for a unicyclic graph.

Keywords: unicyclic graphs; dim( G ); edim( G ) (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/19/3539/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/19/3539/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:19:p:3539-:d:928228

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().