Unit Hypercube Visibility Numbers of Trees

Abstract

A visibility representation of a graph G is an assignment of the vertices of G to geometric objects such that vertices are adjacent if and only if their corresponding objects are “visible” each other, that is, there is an uninterrupted channel, usually axis-aligned, between them. Depending on the objects and definition of visibility used, not all graphs are visibility graphs. In such situations, one may be able to obtain a visibility representation of a graph G by allowing vertices to be assigned to more than one object. The visibility number of a graph G is the minimum t such that G has a representation in which each vertex is assigned to at most t objects. In this paper, we explore visibility numbers of trees when the vertices are assigned to unit hypercubes in $\mathbb {R}^n$. We use two different models of visibility: when lines of sight can be parallel to any standard basis vector of $\mathbb {R}^n$, and when lines of sight are only parallel to the nth standard basis vector in $\mathbb {R}^n$. We establish relationships between these visibility models and their connection to trees with certain cubicity values.