Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Articles in Press 2025 02 27 On metric dimension of cube of trees 14914 10.22049/cco.2025.29750.2143 EN Sanchita Paul Department of Mathematics, C.V. Raman Global University, Bhubaneswar 752054, India Bapan Das Department of Mathematics, Balurghat College, Balurghat 733101, India Avishek Adhikari Department of Mathematics, Presidency University, Kolkata 700073, India Laxman Saha Department of Mathematics, Balurghat College, Balurghat 733101, India Journal Article 2024 05 19 Let $G=(V,E)$ be a connected graph and $d_{G}(u,v)$ be the shortest distance between the vertices $u$ and $v$ in $G$. A set $S=\{s_{1},s_{2},\dots,s_{n}\}\subset V(G)$ is said to be a {\em resolving set} if for all distinct vertices $u,v$ of $G$, there exists an element $s\in S$ such that $d_{G}(s,u)\neq d_{G}(s,v)$. The minimum cardinality of a resolving set for a graph $G$ is called the metric dimension of $G$, and it is denoted by $\beta{(G)}$. A resolving set having $\beta{(G)}$ number of vertices is named as metric basis of $G$. The metric dimension problem is to find a metric basis in a graph $G$, and it has several real-life applications in network theory, telecommunication, image processing, pattern recognition, and many other fields. In this article, we consider cube of trees $T^{3}=(V, E)$, where any two vertices $u,v$ are adjacent if and only if the distance between them is less than or equal to three in $T$. We establish the necessary and sufficient conditions for a vertex subset of $V$ to become a resolving set for $T^{3}$. This helps to determine the tight bounds (upper and lower) on the metric dimension of $T^{3}$. Then, for certain well-known cube of trees, such as caterpillars, lobsters, spiders, and $d$-regular trees, we establish the boundaries for the metric dimension. Also, for every positive integer, we provide a construction showing the existence of a cube of a tree satisfying its metric dimension as the given integer. Further, we characterize some restricted families of cube of trees satisfying $\beta{(T^{3})}=\beta{(T)}$. https://comb-opt.azaruniv.ac.ir/article_14914_8aa3a0a626f18eaee0fe59f8ae89ccf4.pdf