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

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.