Upper bounds for bar visibility of subgraphs and n-vertex graphs

TL;DR

This paper investigates the bar visibility number of graphs, establishing upper bounds and relationships between a graph and its spanning subgraphs, leading to improved bounds for n-vertex graphs.

Contribution

It proves that the bar visibility number of a spanning subgraph is at most one more than that of the original graph, and provides a tighter upper bound for n-vertex graphs.

Findings

If H is a spanning subgraph of G, then b(H) ≤ b(G) + 1.

For an n-vertex graph G, b(G) ≤ ⌈n/6⌉ + 1.

The new bounds improve previous results by Chang et al.

Abstract

A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive width. The least $t$ such that $G$ has a $t$-bar visibility representation is the bar visibility number of $G$, denoted by $b(G)$. We show that if $H$ is a spanning subgraph of $G$, then $b(H)\le b(G)+1$. It follows that $b(G)\le \lceil n/6\rceil+1$ when $G$ is an $n$-vertex graph. This improves the upper bound obtained by Chang et al. (SIAM J. Discrete Math. 18 (2004) 462).