Unit Hypercube Visibility Numbers of Trees

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 Rn. We use two different models of visibility: when lines of sight can be parallel to any standard basis vector of Rn, and when lines of sight are only parallel to the nth standard basis vector in Rn. We establish relationships between these visibility models and their connection to trees with certain cubicity values.