On the SIG-Dimension of Trees Under the L _∞-Metric

Abstract

Let ${{\mathcal P},}$ where ${|{\mathcal P}| \geq 2,}$ be a set of points in d-dimensional space with a given metric ρ. For a point ${p \in {\mathcal P},}$ let r _p be the distance of p with respect to ρ from its nearest neighbor in ${{\mathcal P}.}$ Let B(p,r _p ) be the open ball with respect to ρ centered at p and having the radius r _p . We define the sphere-of-influence graph (SIG) of ${{\mathcal P}}$ as the intersection graph of the family of sets ${\{B(p,r_p)\ | \ p\in {\mathcal P}\}.}$ Given a graph G, a set of points ${{\mathcal P}_G}$ in d-dimensional space with the metric ρ is called a d-dimensional SIG-representation of G, if G is isomorphic to the SIG of ${{\mathcal P}_G.}$ It is known that the absence of isolated vertices is a necessary and sufficient condition for a graph to have a SIG-representation under the L _∞-metric in some space of finite dimension. The SIG-dimension under the L _∞-metric of a graph G without isolated vertices is defined to be the minimum positive integer d such that G has a d-dimensional SIG-representation under the L _∞-metric. It is denoted by SIG _∞(G). We study the SIG-dimension of trees under the L _∞-metric and almost completely answer an open problem posed by Michael and Quint (Discrete Appl Math 127:447–460, 2003). Let T be a tree with at least two vertices. For each ${v\in V(T),}$ let leaf-degree(v) denote the number of neighbors 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) = α}. If |S| = 1, we define β(T) = α(T) − 1. Otherwise define β(T) = α(T). We show that for a tree ${T, SIG_\infty(T) = \lceil \log_2(\beta + 2)\rceil}$ where β = β (T), provided β is not of the form 2^k − 1, for some positive integer k ≥ 1. If β = 2^k − 1, then ${SIG_\infty (T) \in \{k, k+1\}.}$ We show that both values are possible.