On metric dimension of cube of trees

Abstract

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 $S=\{s_{1},s_{2},\cdots,s_{n}\}\subset V(G)$ is said to be a {\em resolving set} if for all distinct vertices $u,v$ of $G$, there exist an element $s\in S$ such that $d(s,u)\neq d(s,v)$. The minimum cardinality of a resolving set for a graph $G$ is called the {\em metric dimension} of $G$ and it is denoted by $β{(G)}$. A resolving set having $β{(G)}$ number of vertices is named as {\em 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 {\em 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 equal to three in $T$. We establish the necessary and sufficient conditions of a vertex subset of $V$ to become a resolving set for $T^{3}$. This helps determine the tight bounds (upper and lower) for the metric dimension of $T^{3}$. Then, for certain well-known cubes of trees, such as caterpillars, lobsters, spiders, and $d$-regular trees, we establish the boundaries of the metric dimension. Further, we characterize some restricted families of cube of trees satisfying $β{(T^{3})}=β{(T)}$. We provide a construction showing the existence of a cube of tree attaining every positive integer value as their metric dimension.

Figures (9)

• Figure 1: Resolvability conditions in $T^{3}$ depending on $d_{T}(u,v)$ (all possible positions of $x$ that resolves $u,v$ are depicted by red vertices)
• Figure 2: For the trees $T_{i}$, $1\leq i\leq 5$ and $S_{i}$ (set of all red vertices), all the conditions of Theorem \ref{['neccsuff']} hold true except condition $i$, which fails for the pair $u,v$ satisfying $d_{T}(u,v)=i$
• Figure 3: Tree $T$ having red vertices as elements of a metric basis of it, blue vertices are extra inserted to form a metric basis $S$ of $T^{3}$, above (left and right) figures correspond to the situation when $T$ contains at least one major stem, and below figures indicate the situation when there is no major stem containing long legs in $T$
• Figure 4: Tree $T$ with $\beta{(T^{3})}=n$ (left when $n$ is odd, right when $n$ is even) where the red vertices form the metric basis of $T$ and the blue vertices are extra inserted to form a metric basis of $T^{3}$
• Figure 5: caterpillar

Theorems & Definitions (45)

• Definition 2.1
• Theorem 2.3
• Theorem 2.4
• Corollary 2.5
• Definition 2.6
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3