Mutual-visibility of the disjointness graph of segments in ${\mathbb R}^2$

TL;DR

This paper investigates the mutual-visibility number of the disjointness graph of segments formed by points in the plane, establishing bounds and properties related to its diameter.

Contribution

It provides tight bounds for the mutual-visibility number of disjointness graphs of segments in the plane and analyzes their diameter properties.

Findings

Established tight bounds for the mutual-visibility number 5(G)

Showed that almost all edge disjointness graphs have diameter 2

Analyzed properties of disjointness graphs in 5^2

Abstract

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 $\mu(G)$ 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 $\mu(D(P))$, and show that almost all edge disjointness graphs have diameter 2.