On the SIG dimension of trees under $L_{\infty}$ metric

TL;DR

This paper determines the SIG dimension of trees under the $L_{}$ metric, providing a formula based on leaf-degree properties and resolving an open problem from prior research.

Contribution

It establishes an explicit formula for the SIG dimension of trees under $L_{}$ metric, addressing an open problem and characterizing cases with specific $eta$ values.

Findings

SIG_(T) = (eta + 2) for most trees

For special (eta) = 2^k - 1, SIG_(T) is either k or k+1

Both possible values occur for the case (eta) = 2^k - 1

Abstract

We study the $SIG$ dimension of trees under $L_{\infty}$ metric and answer an open problem posed by Michael and Quint (Discrete Applied Mathematics: 127, pages 447-460, 2003). Let $T$ be a tree with atleast two vertices. For each $v\in V(T)$, let leaf-degree$(v)$ denote the number of neighbours of $v$ that are leaves. We define the maximum leaf-degree as $\alpha(T) = \max_{x \in V(T)}$ leaf-degree$(x)$. Let $S = \{v\in V(T) |$ leaf-degree$(v) = \alpha\}$. If $|S| = 1$, we define $\beta(T) = \alpha(T) - 1$. Otherwise define $\beta(T) = \alpha(T)$. We show that for a tree $T$, $SIG_\infty(T) = \lceil \log_2(\beta + 2)\rceil$ where $\beta = \beta (T)$, provided $\beta$ is not of the form $2^k - 1$, for some positive integer $k \geq 1$. If $\beta = 2^k - 1$, then $SIG_\infty (T) \in \{k, k+1\}$. We show that both values are possible.