Minimax Theorems for Normal Hypergraphs and Balanced Hypergraphs — A Survey

Abstract

This chapter discusses the collection of minimax properties for normal hypergraphs and balanced hypergraphs. A hypergraph H is a family of non-empty subsets, called “edges.” A set T ⊂ X is a transversal set of H if T meets all the edges; the family of all the minimal transversal sets is called the “transversal hypergraph” and is denoted by TrH. Let G be a directed graph, and let A be a vertex. Consider the hypergraph H whose vertices are the arcs of G, whose edges are spanning arborescences rooted in a. A hypergraph satisfies the Gupta property only if its transversal hypergraph satisfies the Menger property. It is easy to show that the transversal hypergraph of the T-join hypergraph is the T-cut hypergraph, and vice versa.