Caterpillars and alternating paths

TL;DR

This paper derives explicit formulas for the maximum size of caterpillars obtainable from any tree with a given number of edges through contraction or vertex removal, linking these to geometric interpretations involving alternating paths.

Contribution

It provides closed-form expressions for functions related to transforming trees into caterpillars and connects these to geometric problems involving alternating paths among line segments.

Findings

Explicit formulas for p(m) and q(m) for all m ≥ 1.

Connections between tree transformations and geometric alternating path problems.

Insights into the structure of caterpillars derived from arbitrary trees.

Abstract

Let $p(m)$ (respectively, $q(m)$) be the maximum number $k$ such that any tree with $m$ edges can be transformed by contracting edges (respectively, by removing vertices) into a caterpillar with $k$ edges. We derive closed-form expressions for $p(m)$ and $q(m)$ for all $m \ge 1$. The two functions $p(n)$ and $q(n)$ can also be interpreted in terms of alternating paths among $n$ disjoint line segments in the plane, whose $2n$ endpoints are in convex position.