1-Visibility Representations of 1-Planar Graphs

TL;DR

This paper introduces 1-visibility representations for 1-planar graphs, extending classical visibility representations, and provides a linear-time algorithm for their construction, revealing new insights into graph density and relationships.

Contribution

It generalizes classical visibility representations to 1-visibility, extending the concept to 1-planar graphs, and offers a linear-time algorithm for constructing these representations.

Findings

1-visibility extends 1-planarity.

Both 1-visible and 1-planar graphs have at most 4n-8 edges.

There exist 1-visible graphs with 4n-8 edges that are not 1-planar.

Abstract

A visibility representation is a classical drawing style of planar graphs. It displays the vertices of a graph as horizontal vertex-segments, and each edge is represented by a vertical edge-segment touching the segments of its end vertices; beyond that segments do not intersect. We generalize visibility to 1-visibility, where each edge- (vertex-) segment crosses at most one vertex- (edge-) segment. In other words, a vertex is crossed by at most one edge, and vice-versa. We show that 1-visibility properly extends 1-planarity and develop a linear time algorithm to compute a 1-visibility representation of an embedded 1-planar graph on O(n^2) area. A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. Concerning density, both 1-visible and 1-planar graphs of size $n$ have at most 4n-8 edges. However, for every n >= 7 there are 1-visible graphs with 4n-8 edge which are not 1-planar.