
TL;DR
This paper investigates the minimum number of total orderings needed to satisfy certain betweenness and nonbetweenness conditions among objects, providing exact and asymptotic formulas and connecting these to phylogenetic tree structures.
Contribution
It derives exact and asymptotic formulas for betweenness and nonbetweenness functions, and introduces the extreme ternary constraint function as a generalization.
Findings
nbet(n) = ⌈log₂log₂n⌉ + 1
bet(n) = Θ(log n)
Minimum size of rooted phylogenetic trees set is Θ(log log n)
Abstract
The betweenness function is the minimum number of total orderings of objects such that for any three distinct objects , and , there is an ordering in which is between and . The nonbetweenness function is the minimum number of total orderings such that for any three distinct objects , and , there is an ordering in which is not between and . We show that and . Betweenness and Nonbetweenness are specific cases of a more general extreme value function called the `extreme ternary constraint function'. The asymptotic value of this generalisation is computed using the values of and . This result demonstrates that the minimum size of a set of rooted phylogenetic trees is consistent with all phylogenetic triplets is .
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Taxonomy
TopicsAdvanced Graph Theory Research
