The Tree Stabbing Number is not Monotone

TL;DR

This paper demonstrates that the tree stabbing number, a measure of how a spanning tree intersects lines, is not monotonic with respect to point set inclusion, answering an open problem in computational geometry.

Contribution

It proves that the tree stabbing number is not a monotone parameter, providing a counterexample to a previously open question.

Findings

Tree stabbing number is not monotone with point set inclusion

Counterexample disproves monotonicity of the parameter

Answers an open problem in computational geometry

Abstract

Let $P \subseteq \mathbb{R}^2$ be a set of points and $T$ be a spanning tree of $P$. The \emph{stabbing number} of $T$ is the maximum number of intersections any line in the plane determines with the edges of $T$. The \emph{tree stabbing number} of $P$ is the minimum stabbing number of any spanning tree of $P$. We prove that the tree stabbing number is not a monotone parameter, i.e., there exist point sets $P \subsetneq P'$ such that \treestab{$P$} $>$ \treestab{$P'$}, answering a question by Eppstein \cite[Open Problem~17.5]{eppstein_2018}.