Sharp upper and lower bounds on a restricted class of convex characters

TL;DR

This paper establishes sharp bounds on the number of convex characters with blocks of size at least k in unrooted binary trees, revealing how tree topology influences these counts and exploring potential algorithmic uses.

Contribution

It characterizes tree topologies that maximize and minimize the number of convex characters with size constraints for all k ≥ 3, extending previous work on k=1,2.

Findings

Identifies trees with extremal counts of convex characters for k ≥ 3.

Provides explicit formulas and exponential bounds for g_k.

Highlights potential algorithmic applications of these bounds.

Abstract

Let $\mathcal{T}$ be an unrooted binary tree with $n$ distinctly labelled leaves. Deriving its name from the field of phylogenetics, a convex character on $\mathcal{T}$ is simply a partition of the leaves such that the minimal spanning subtrees induced by the blocks of the partition are mutually disjoint. In earlier work Kelk and Stamoulis (Advances in Applied Mathematics 84 (2017), pp. 34--46) defined $g_k(\mathcal{T})$ as the number of convex characters where each block has at least $k$ leaves. Exact expressions were given for $g_1$ and $g_2$, where the topology of $\mathcal{T}$ turns out to be irrelevant, and it was noted that for $k \geq 3$ topological neutrality no longer holds. In this article, for every $k \geq 3$ we describe tree topologies achieving the maximum and minimum values of $g_k$ and determine corresponding expressions and exponential bounds for $g_k$. Finally, we reflect briefly on possible algorithmic applications of these results.