Convex geometric (k+2)-quasiplanar representations of semi-bar k-visibility graphs

TL;DR

This paper studies semi-bar visibility graphs, demonstrating their (k+2)-quasiplanar representations in the plane and characterizing those with cylindrical semi-bar representations as (2k+2)-degenerate graphs with maximal (k+2)-quasiplanar representations.

Contribution

It establishes that semi-bar k-visibility graphs have (k+2)-quasiplanar representations and characterizes cylindrical semi-bar k-visibility graphs as (2k+2)-degenerate graphs with maximal representations.

Findings

Semi-bar k-visibility graphs have (k+2)-quasiplanar representations.

Cylindrical semi-bar k-visibility graphs correspond to (2k+2)-degenerate graphs.

Maximal (k+2)-quasiplanar representations are characterized for cylindrical semi-bar graphs.

Abstract

We examine semi-bar visibility graphs in the plane and on a cylinder in which sightlines can pass through k objects. We show every semi-bar k-visibility graph has a (k+2)-quasiplanar representation in the plane with vertices drawn as points in convex position and edges drawn as segments. We also show that the graphs having cylindrical semi-bar k-visibility representations with semi-bars of different lengths are the same as the (2k+2)-degenerate graphs having edge-maximal (k+2)-quasiplanar representations in the plane with vertices drawn as points in convex position and edges drawn as segments.