The Unit Bar Visibility Number of a Graph

TL;DR

This paper introduces the concept of the unit bar visibility number of a graph, providing algorithms and bounds for specific classes such as trees and complete bipartite graphs, advancing understanding of graph visibility representations.

Contribution

It defines the unit bar visibility number and offers a linear time algorithm for trees, along with bounds and exact values for complete bipartite and complete graphs.

Findings

Linear time algorithm for trees' unit bar visibility number

Bounds on $ub(K_{m,n})$ for asymptotic cases

Exact $ub(K_n)$ values for certain $n$ modulo 6

Abstract

A \textit{$t$-unit-bar representation} of a graph $G$ is an assignment of sets of at most $t$ horizontal unit-length segments in the plane to the vertices of $G$ so that (1) all of the segments are pairwise nonintersecting, and (2) two vertices $x$ and $y$ are adjacent if and only if there is a vertical channel of positive width connecting a segment assigned to $x$ and a segment assigned to $y$ that intersects no other segment. The \textit{unit bar visibility number} of a graph $G$, denoted $ub(G)$, is the minimum $t$ such that $G$ has a $t$-unit-bar visibility representation. Our results include a linear time algorithm that determines $ub(T)$ when $T$ is a tree, bounds on $ub(K_{m,n})$ that determine $ub(K_{m,n})$ asymptotically when $n$ and $m$ are asymptotically equal, and bounds on $ub(K_n)$ that determine $ub(K_n)$ exactly when $n\equiv 1,2\pmod 6$.