τ–Critical Hypergraphs and the Helly Property

Abstract

Publisher Summary An early result of Gilmore can be formulated as a sufficient (and necessary) condition on the “local” structure of a hypergraph having the Helly property. Berge and Duchet have extended Gilmore's condition to hypergraphs having the Helly property of “higher dimension.” A hypergraph is said to be τ-critical if the removal of any edge reduces the transversal number. If the transversal number of a τ-critical hypergraph H is known, say τ(H)= t + 1, then H will be called “( t + 1)-critical.” A hypergraph is called “k –intersecting” if it contains at least k edges and any k of its edges have a nonempty intersection. A hypergraph H verifies the (k , t)-property if τ(H′) ≤ t holds for every k-intersecting partial hypergraph H’ of H. The chapter presents that the analogous characterizations of hypergraphs having the (k, t)-property follow from the solution of extremal problems posed on k - intersecting (t + 1)-critical hypergraphs.