THE METRIC DIMENSION OF CYCLE BOOK GRAPHS B_cm,nFORMED BY A COMMON PATH P₂

This paper investigates the metric dimension of a class of graphs known as cycle books, denoted B_cm,nwhich feature a shared path P₂across 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 dim(B_c3n)=n for n≥2, and dim(B_c4n)=n for n=2,3,4, while dim(B_c4n)=n−1 for n≥5. Furthermore, we provide a general result for m≥5: the metric dimension is n when m is odd and m∈{2,3}, or when m is even and n≥2; and n−1 when m is odd and n≥4. These findings contribute to the growing body of knowledge on metric properties in graph theory, particularly in structured and cyclic graph families.