Abstract
Let V be a countable set, let T be a rooted tree on the vertex set V, and let \({\mathcal M}=(V,2^V, \mu )\) be a finite signed measure space. How can we describe the “shape” of the weighted rooted tree \((T, {\mathcal M})\)? Is there a natural criterion for calling it “fat” or “tall”? We provide a series of such criteria and show that every “heavy” weighted rooted tree is either fat or tall, as we wish. This leads us to seek hypergraphs such that regardless of how we assign a finite signed measure on their vertex sets, the resulting weighted hypergraphs have either a “heavy” large matching or a “heavy” vertex subset that induces a subhypergraph with small matching number. Here we also must develop an appropriate definition of what it means for a set to be heavy in a signed measure space.
Supported by NSFC (11671258, 11971305) and STCSM (17690740800).
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More precisely, an Alexandroff space is the set of down-sets of a preorder.
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Wu, Y., Zhu, Y. (2020). Weighted Rooted Trees: Fat or Tall?. In: Fernau, H. (eds) Computer Science – Theory and Applications. CSR 2020. Lecture Notes in Computer Science(), vol 12159. Springer, Cham. https://doi.org/10.1007/978-3-030-50026-9_30
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DOI: https://doi.org/10.1007/978-3-030-50026-9_30
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