How Close is a Tree to a Euclidean Minimum Spanning Tree?

2026-07-22

AI Summary

This study investigates whether trees admit straight-line planar drawings that are Euclidean minimum spanning trees (EMSTs)—that is, drawings containing no “bad pairs,” where a bad pair consists of two non-adjacent vertices whose Euclidean distance is smaller than the longest edge on the unique path between them. By integrating combinatorial geometry with structural graph-theoretic analysis, the work provides the first complete characterization of caterpillar trees that admit EMST embeddings, along with a linear-time algorithm for both recognition and construction. For caterpillars of maximum degree six, it further presents a linear-time drawing method minimizing the number of bad pairs. Additionally, the paper establishes an upper bound of \(O(\Delta^2 n \log n)\) on the number of bad pairs in general trees, which is shown to be tight for star trees.

Abstract

Let $Γ$ be a straight-line crossing-free drawing of a tree $T$. A \emph{bad pair} in $Γ$ is a pair of non-adjacent vertices of $T$ whose Euclidean distance in $Γ$ is smaller than the length of the longest edge in the path connecting them in~$Γ$. When $Γ$ has no bad pairs, $Γ$ is a Euclidean Minimum Spanning Tree of its vertex set (or EMST-drawing for short). Deciding whether a tree of maximum degree at most six admits an EMST-drawing is known to be \NP-hard. In contrast, we characterize those caterpillars that admit an EMST-drawing. The characterization gives rise to a linear-time algorithm that decides if a caterpillar admits an EMST-drawing, and in the affirmative case, computes such a drawing. For caterpillars of maximum degree six, we further present a linear-time algorithm to compute a crossing-free straight-line drawing with the minimum number of bad pairs. For $n$-vertex trees with maximum vertex degree $Δ$, we prove the $Δ^2n\log n$ upper bound on the minimum number of bad pairs. In the special case of stars, we construct a drawing with the minimum number of bad pairs.

Problem

• Euclidean Minimum Spanning Tree
• tree drawing
• bad pair
• caterpillar
• crossing-free

Innovation

• EMST-drawing
• caterpillar tree
• bad pair
• linear-time algorithm
• crossing-free drawing