Equality of domination and transversal numbers in hypergraphs

S. Arumugam, Bibin K. Jose, Csilla Bujtás, Zsolt Tuza

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

A subset S of the vertex set of a hypergraph H is called a dominating set of H if for every vertex v not in S there exists u ε S such that u and v are contained in an edge in H. The minimum cardinality of a dominating set in H is called the domination number of H and is denoted by γ(H). A transversal of a hypergraph H is defined to be a subset T of the vertex set such that Ta ∩ E θ for every edge E of H. The transversal number of H, denoted by τ(H), is the minimum number of vertices in a transversal. A hypergraph is of rank k if each of its edges contains at most k vertices. The inequality τ(H)≥γ(H) is valid for every hypergraph H without isolated vertices. In this paper, we investigate the hypergraphs satisfying τ(H)=γ(H), and prove that their recognition problem is NP-hard already on the class of linear hypergraphs of rank 3, while on unrestricted problem instances it lies inside the complexity class Θ2p. Structurally we focus our attention on hypergraphs in which each subhypergraph H′ without isolated vertices fulfills the equality τ(H′)=γ(H′). We show that if each induced subhypergraph satisfies the equality then it holds for the non-induced ones as well. Moreover, we prove that for every positive integer k, there are only a finite number of forbidden subhypergraphs of rank k, and each of them has domination number at most k.

Original languageEnglish
Pages (from-to)1859-1867
Number of pages9
JournalDiscrete Applied Mathematics
Volume161
Issue number13-14
DOIs
Publication statusPublished - Sep 1 2013

Keywords

  • Computational complexity
  • Domination number
  • Hereditary property
  • Hitting set
  • Hypergraph
  • Polynomial hierarchy
  • Transversal number

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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