Pohon dengan bilangan kromatik-lokasi dan dimensi partisi mendominasi sama dengan tiga - Muhammad Ridwan; Abidin, Wahyuni (Jurnal MSA (Matematika dan Statistika serta Aplikasinya), 2025)
Several graph parameters are defined through vertex partitions and distance representations. Two such parameters are the locating-chromatic number $χ_L(G)$ and the dominating partition dimension $η_p(G)$. The locating-chromatic number requires that vertices are distinguished by their distances to color classes, while the dominating partition dimension additionally imposes a domination condition, namely that every vertex must be adjacent to at least one partition class. It is known that for every connected graph $G$, $β_p(G)\le η_p(G)\le χ_L(G)$, where $β_p(G)$ denotes the partition dimension. However, the conditions under which equality holds are not yet fully understood. In this paper, we focus on trees $T$ and show that the equality $χ_L(T)=η_p(T)$ occurs exactly at value three. More precisely, we prove that a tree $T$ satisfies $χ_L(T)=3$ if and only if $η_p(T)=3$. The proof is structural and is based on an analysis of leaf configurations and strong support vertices. In particular, we establish that the presence of two distinct strong support vertices forces $η_p(T)\ge 4$, preventing equality at value three. As a consequence, the class of trees with locating-chromatic number three coincides exactly with the class of trees whose dominating partition dimension equals three.