Equality of domination and transversal numbers in hypergraphs

  • Authors:

  • S. Arumugam;Bibin K. Jose;Csilla BujtáS;Zsolt Tuza

  • Affiliations:

  • National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH), Kalasalingam University, Anand Nagar, Krishnankoil-626126, India and School of Electrical Engineering and Computer Scien ...;PG Department of Mathematics, S. D. College Alappuzha, Sanathanapuram P.O., Aleppey-688003, Kerala, India;Department of Computer Science and Systems Technology, University of Pannonia, H-8200 Veszprém, Egyetem u. 10, Hungary;Department of Computer Science and Systems Technology, University of Pannonia, H-8200 Veszprém, Egyetem u. 10, Hungary and Alfréd Rényi Institute of Mathematics, Hungarian Academy o ...

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2013

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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@?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 @c(H). A transversal of a hypergraph H is defined to be a subset T of the vertex set such that T@?E0@? for every edge E of H. The transversal number of H, denoted by @t(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 @t(H)=@c(H) is valid for every hypergraph H without isolated vertices. In this paper, we investigate the hypergraphs satisfying @t(H)=@c(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 @Q"2^p. Structurally we focus our attention on hypergraphs in which each subhypergraph H^' without isolated vertices fulfills the equality @t(H^')=@c(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.