EXACT OPEN-LOCATING-DOMINATING NUMBERS OF HELM, CYCLE CORONA, AND GEAR GRAPHS

Abstract

An open-locating-dominating (OLD) set in a graph is a subset of vertices that dominates the graph and distinguishes all pairs of vertices via their open neighborhoods within the set. In this paper, we determine the exact OLD number $\gamma_{\mathrm{OLD}}(G)$ for three families of structured graphs. We first establish that for helm graphs $H_n$ and cycle coronae $C_n \circ K_1$, the parameter equals $n$ for all $n \ge 3$, except for $n=4$ where $\gamma_{\mathrm{OLD}}(H_4)=6$. We then focus on gear graphs $G_n$, where standard degree-based bounds prove insufficient. By modeling the problem via binary sequences and analyzing forbidden local patterns, we prove that for $n \ge 4$, $\gamma_{\mathrm{OLD}}(G_n) = 2\lceil 2n/3\rceil$, and $\gamma_{\mathrm{OLD}}(G_3)=5$.