A min-max relation for stable sets in graphs with no odd- K4

Abstract

We prove the following min-max relations. Let G be an undirected graph, without isolated nodes, not containing an odd- K 4 (a homeomorph of K 4 (the complete graph with four nodes) in which the triangles of K 4 have become odd circuits). Then the maximum cardinality of a stable set in G is equal to the minimum cost of a collection of edges and odd circuits in G, covering the nodes of G. Here the cost of an edge is 1 and the cost of a circuit of length 2 k + 1 is equal to k. Moreover, the minimum cardinality of a node-cover for G is equal to the maximum profit of a collection of mutually node disioint edges and odd circuits in G. Here the profit of an edge is 1 and the profit of a circuit of length 2 k + 1 is equal to k + 1. Also, weighted versions of these min-max relations hold. The result extends König's well-known min-max relations for stable sets and node-covers in bipartite graphs. It also extends results of Chvátal, Boulala, Fonlupt, and Uhry. A weaker, fractional, version of these min-max relations follows from earlier results obtained by Schrijver and the author.