THE METRIC DIMENSION OF CYCLE BOOK GRAPHS B_(C_(m,n) ) FORMED BY A COMMON PATH P_2

  • Jaya Santoso Department of Informatics, Faculty of Informatics and Electrical Engineering, Institut Teknologi Del, Indonesia https://orcid.org/0009-0001-7329-3137
  • Darmaji Darmaji 2Department of Mathematics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember Gedung G Lantai 2, Indonesia https://orcid.org/0009-0000-5213-1334
  • Ana Muliyana Software Engineering Technology, Faculty of Vocational, Institut Teknologi Del, Indonesia https://orcid.org/0009-0004-6304-0058
  • Asido Saragih Department of Electrical Engineering, Faculty of Informatics and Electrical Engineering, Institut Teknologi Del, Indonesia https://orcid.org/0009-0007-8851-3490
DOI: https://doi.org/10.30598/barekengvol20iss2pp1155–1166
Keywords: Cycle books graph, Graph structures, Graph theory, Metric dimension, Resolving set

Abstract

This paper investigates the metric dimension of a class of graphs known as cycle books, denoted ​, which feature a shared path ​ across multiple cycles. We focus on characterizing the minimum number of vertex subsets required so that each vertex in the graph can be uniquely identified by its distances to those subsets. To support our analysis, we present two propositions and a general theorem that establish the metric dimension for various configurations of cycle book graphs. Specifically, we prove that  for , and  for , while  for . Furthermore, we provide a general result for : the metric dimension is  when  is odd and , or when  is even and ; and  when  is odd and . These findings contribute to the growing body of knowledge on metric properties in graph theory, particularly in structured and cyclic graph families.This paper investigates the metric dimension of a class of graphs known as cycle books, denoted ​, which feature a shared path ​ across multiple cycles. We focus on characterizing the minimum number of vertex subsets required so that each vertex in the graph can be uniquely identified by its distances to those subsets. To support our analysis, we present two propositions and a general theorem that establish the metric dimension for various configurations of cycle book graphs. Specifically, we prove that  for , and  for , while  for . Furthermore, we provide a general result for : the metric dimension is  when  is odd and , or when  is even and ; and  when  is odd and . These findings contribute to the growing body of knowledge on metric properties in graph theory, particularly in structured and cyclic graph families.This paper investigates the metric dimension of a class of graphs known as cycle books, denoted ​, which feature a shared path ​ across multiple cycles. We focus on characterizing the minimum number of vertex subsets required so that each vertex in the graph can be uniquely identified by its distances to those subsets. To support our analysis, we present two propositions and a general theorem that establish the metric dimension for various configurations of cycle book graphs. Specifically, we prove that  for , and  for , while  for . Furthermore, we provide a general result for : the metric dimension is  when  is odd and , or when  is even and ; and  when  is odd and . These findings contribute to the growing body of knowledge on metric properties in graph theory, particularly in structured and cyclic graph families.

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Published
2026-01-26
How to Cite
[1]
J. Santoso, D. Darmaji, A. Muliyana, and A. Saragih, “THE METRIC DIMENSION OF CYCLE BOOK GRAPHS B_(C_(m,n) ) FORMED BY A COMMON PATH P_2”, BAREKENG: J. Math. & App., vol. 20, no. 2, pp. 1155–1166, Jan. 2026.
Issue
Vol 20 No 2 (2026): BAREKENG: Journal of Mathematics and Its Application
Section
Articles

Copyright (c) 2026 Jaya Santoso, Darmaji Darmaji, Ana Muliyana, Asido Saragih

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