Bar 1-Visibility Drawings of 1-Planar Graphs

TL;DR

This paper introduces linear-time algorithms for creating bar 1-visibility drawings of certain 1-planar graphs, expanding understanding of graph visualization techniques for these classes.

Contribution

It provides the first linear-time algorithms for bar 1-visibility drawings of diagonal grid and maximal outer 1-planar graphs, and proves other classes are bar 1-visible.

Findings

Linear-time algorithms for diagonal grid graphs and maximal outer 1-planar graphs.

Recursive quadrangle and pseudo double wheel 1-planar graphs are bar 1-visible.

Enhanced understanding of graph visualization for 1-planar graphs.

Abstract

A bar 1-visibility drawing of a graph $G$ is a drawing of $G$ where each vertex is drawn as a horizontal line segment called a bar, each edge is drawn as a vertical line segment where the vertical line segment representing an edge must connect the horizontal line segments representing the end vertices and a vertical line segment corresponding to an edge intersects at most one bar which is not an end point of the edge. A graph $G$ is bar 1-visible if $G$ has a bar 1-visibility drawing. A graph $G$ is 1-planar if $G$ has a drawing in a 2-dimensional plane such that an edge crosses at most one other edge. In this paper we give linear-time algorithms to find bar 1-visibility drawings of diagonal grid graphs and maximal outer 1-planar graphs. We also show that recursive quadrangle 1-planar graphs and pseudo double wheel 1-planar graphs are bar 1-visible graphs.