Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.00689 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Let $G=(V(G),E(G))$ be a simple graph, and let $U\subseteq V(G)$. Two distinct vertices $x,y\in U$ are $U$-mutually visible if $G$ contains a shortest $x$-$y$ path that is internally disjoint from $U$. $U$ is called a mutual-visibility set of $G$ if any two vertices of $U$ are $U$-mutually visible. The mutual-visibility number $μ(G)$ of $G$ is the size of a largest mutual-visibility set of $G$. Let $P$ be a set of $n\geq 3$ points in ${\mathbb R}^2$ in general position. The disjointness graph of segments $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in $D(P)$ if and only if they are disjoint. In this paper we establish tight lower and upper bounds for $μ(D(P))$, and show that almost all edge disjointness graphs have diameter 2.