Unit Hypercube Visibility Numbers of Trees

TL;DR

This paper investigates the minimum number of unit hypercubes needed to represent trees with visibility between hypercubes in different models, linking these numbers to cubicity values of trees.

Contribution

It introduces and analyzes the concept of visibility numbers of trees with unit hypercubes in multiple models, establishing relationships with cubicity.

Findings

Visibility numbers depend on the model of line-of-sight considered.

Connections between visibility numbers and cubicity of trees are established.

Results provide bounds and relationships for hypercube visibility representations.

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 {\it 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 $n$th standard basis vector in $\mathbb{R}^n$. We establish relationships between these visibility models and their connection to trees with certain cubicity values.