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

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 $n\ge 3$ 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.

19 pages, 5 figures