Further results on arc and bar k-visibility graphs

TL;DR

This paper advances the understanding of arc and bar k-visibility graphs by establishing new bounds, characterizations, and models, enhancing the theoretical framework and exploring their properties and relationships.

Contribution

It provides new edge bounds, maximal constructions, complete characterizations, and introduces a random graph model for arc and semi-arc k-visibility graphs.

Findings

New edge bounds for arc k-visibility graphs

Complete characterization of semi-arc visibility graphs

Introduction and analysis of a random semi-arc k-visibility graph model

Abstract

We consider visibility graphs involving bars and arcs in which lines of sight can pass through up to k objects. We prove a new edge bound for arc k-visibility graphs, provide maximal constructions for arc and semi-arc k-visibility graphs, and give a complete characterization of semi-arc visibility graphs. We show that the family of arc i-visibility graphs is never contained in the family of bar j-visibility graphs for any i and j, and that the family of bar i-visibility graphs is not contained in the family of bar j-visibility graphs for $i \neq j$. We also give the first thickness bounds for arc and semi-arc k-visibility graphs. Finally, we introduce a model for random semi-bar and semi-arc k-visibility graphs and analyze its properties.