Heavy Transversals and Indecomposable Hypergraphs

A weighted hypergraph is a hypergraph H = (V, E) with a weighting function \( w:V \to R \), where R is the set of reals. A multiset S ⊆ V generates a partial hypergraph H S with edges \( {\left\{ {e \in E{\kern 1pt} :{\kern 1pt} {\left| {e \cap S} \right|} > w{\left( S \right)}} \right\}} \), where both the cardinality \( {\left| {e \cap S} \right|} \) and the total weight w(S) are counted with multiplicities of vertices in S. The transversal number of H is represented by τ(H). We prove the following: there exists a function f(n) such that, for any weighted n-Helly hypergraph H, τ(H B) ≤ 1, for all multisets BV if and only if τ(H A) ≤ 1, for all multisets AV with \( {\left| A \right|} \leqslant f{\left( n \right)} \). We provide lower and upper bounds for f(n) using a link between indecomposable hypergraphs and critical weighted n-Helly hypergraphs.

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Correspondence to André E. Kézdy.

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* On leave from Computing and Automation Research Institute, Hungarian Academy of Sciences.

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Kézdy, A.E., Lehel*, J. & Powers, R.C. Heavy Transversals and Indecomposable Hypergraphs. Combinatorica 23, 303–310 (2003). https://doi.org/10.1007/s00493-003-0023-2

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AMS Subject Classification (2000):

  • 05C65
  • 05C90
  • 05D15