Drawing a Rooted Tree as a Rooted $y-$Monotone Minimum Spanning Tree

TL;DR

This paper studies the properties of rooted y-monotone minimum spanning trees, providing algorithms for drawing any rooted tree as such and analyzing their area requirements.

Contribution

It introduces a linear time algorithm to draw any rooted tree as a rooted y-monotone minimum spanning tree and demonstrates the unbounded degree and exponential area requirements.

Findings

Maximum degree of rooted y-MMST is unbounded.

Any rooted tree can be drawn as a rooted y-MMST in linear time.

Some rooted trees require exponential grid area for drawing.

Abstract

Given a rooted point set $P$, the rooted $y-$Monotone Minimum Spanning Tree (rooted $y-$MMST) of $P$ is the spanning geometric graph of $P$ in which all the vertices are connected to the root by some $y-$monotone path and the sum of the Euclidean lengths of its edges is the minimum. We show that the maximum degree of a rooted $y-$MMST is not bounded by a constant number. We give a linear time algorithm that draws any rooted tree as a rooted $y-$MMST and also show that there exist rooted trees that can be drawn as rooted $y-$MMSTs only in a grid of exponential area.