On k-visibility graphs

TL;DR

This paper investigates various types of k-visibility graphs, establishing bounds on their thickness, edges, and chromatic number, revealing structural properties and limitations as k varies.

Contribution

It provides new bounds on the thickness, edges, and chromatic number for semi-bar, arc, and circle k-visibility graphs, advancing understanding of their combinatorial properties.

Findings

Bound the maximum thickness of semi-bar k-visibility graphs between rac{2}{3}(k+1) and 2k.

Maximum edges in arc and circle k-visibility graphs are at most (k+1)(3n - k - 2) for large n.

Maximum chromatic number for these graphs is at most 6k+6.

Abstract

We examine several types of visibility graphs in which sightlines can pass through $k$ objects. For $k \geq 1$ we bound the maximum thickness of semi-bar $k$-visibility graphs between $\lceil \frac{2}{3} (k + 1) \rceil$ and $2k$. In addition we show that the maximum number of edges in arc and circle $k$-visibility graphs on $n$ vertices is at most $(k+1)(3n-k-2)$ for $n > 4k+4$ and ${n \choose 2}$ for $n \leq 4k+4$, while the maximum chromatic number is at most $6k+6$. In semi-arc $k$-visibility graphs on $n$ vertices, we show that the maximum number of edges is ${n \choose 2}$ for $n \leq 3k+3$ and at most $(k+1)(2n-\frac{k+2}{2})$ for $n > 3k+3$, while the maximum chromatic number is at most $4k+4$.