A Minimax Equality Related to the Longest Directed Path in an Acyclic Graph

Abstract

As an analog of a recently established minimax equality for directed graphs [1], I. Simon has suggested that the following be investigated.1.1. For a finite acyclic directed graph G, a minimum collection of directed coboundaries whose union is the edge set of G has cardinality equal to that of a maximum strong matching of G.This minimax equality is here proved, using a characterization of a maximum strong matching of an acyclic graph as the set of edges of a longest directed path in the graph.The terms employed in the above theorem are defined as follows. Let G be a finite directed graph with vertex set VG and edge set eG