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

S. Kelk*, R. Meuwese

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

Let T be an unrooted binary tree with n distinctly labelled leaves. Deriving its name from the field of phylogenetics, a convex character on 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)(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 T turns out to be irrelevant, and it was noted that for k >= 3 topological neutrality no longer holds. In this article, for every k >= 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.
Original languageEnglish
Article numberP1.50
Number of pages17
JournalElectronic Journal of Combinatorics
Volume29
Issue number1
DOIs
Publication statusPublished - 11 Mar 2022

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