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arXiv · 0705.1806

Transversals in trees

Victor Campos·Vasek Chvatal·Luc Devroye·Perouz Taslakian
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Abstract

A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n nodes by saying that a tree T succeeds a tree T' if c(T,k) is at least c(T',k) for all k and strictly greater than c(T',k) for at least one k. We prove that, for every choice of positive integers d and n, the set of all rooted trees on n nodes where each node has at most d children has a unique minimal element with respect to this partial order and we describe this tree.

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Victor Campos, Vasek Chvatal, Luc Devroye, Perouz Taslakian. 2007-05-13. Transversals in trees. https://arxiv.org/abs/0705.1806

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