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

Let P , where |P| ≥ 2, be a set of points in d-dimensional space with a given metric ρ. For a point p ∈ P , let rp be the distance of p with respect to ρ from its nearest neighbour in P . Let B(p, rp) be the open ball with respect to ρ centered at p and having the radius rp. We define the sphere-of-influence graph (SIG) of P as the intersection graph of the family of sets {B(p, rp) | p ∈ P}. Given a graph G, a set of points PG in d-dimensional space with the metric ρ is called a d−dimensional SIG representation of G, if G is the SIG of PG. It is known that the absence of isolated vertices is a necessary and sufficient condition for a graph to have an SIG representation under L∞ metric in some space of finite dimension. The SIG dimension under 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 as SIG∞(G). We study the SIG dimension of trees under L∞ 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 ∈ V (T ), let leaf-degree(v) denote the number of neighbours of v that are leaves. We define the maximum leaf-degree as α(T ) = maxx∈V (T ) leaf-degree(x). Let S = {v ∈ 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∞(T ) = ⌈log2(β + 2)⌉ where β = β(T ), provided β is not of the form 2 − 1, for some positive integer k ≥ 1. If β = 2 − 1, then SIG∞(T ) ∈ {k, k + 1}. We show that both values are possible.