The Tree Stabbing Number is not Monotone

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 $P⊊P^{\prime }$ such that \treestab{$P$} $>$ \treestab{$P^{\prime }$}, answering a question by Eppstein \cite[Open Problem~17.5]{eppstein_2018}.