TY  - JOUR
ID  - Leaños2026
LB  - Leaños2026
AU  - Leaños, Jesús
AU  - Lomelí-Haro, Mario
AU  - Ndjatchi, Christophe
AU  - Ríos-Castro, Luis Manuel
TI  - Mutual-visibility of the disjointness graph of segments in R<sup>2</sup>
ST  - Mutual-visibility of the disjointness graph of segments in R<sup>2</sup>
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2026
VL  - 46
IS  - 5
SP  - 679
EP  - 707
DO  - https://doi.org/10.7494/OpMath.202610122
UR  - https://doi.org/10.7494/OpMath.202610122
AB  - 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 \(D(P)\) has diameter 2 for every point set with \(n\) at least 9.
KW  - disjointness graph of segments
KW  - mutual-visibility
KW  - straight-line drawings of complete graphs
SN  - 1232-9274
ER  -