An algorithm for a minimum cover of a graph

Abstract

Introduction. This paper contains two similar theorems giving conditions for a minimum cover and a maximum matching of a graph. Both of these conditions depend on the concept of an alternating path, due to Petersen [2]. These results immediately lead to algorithms for a minimum cover and a maximum matching respectively. The first part of the paper contains a noninductive proof of the minimum cover theorem (Theorem 1) and the resultant algorithm. Next we define a set of on the set of minimum covers; any minimum cover can be obtained from any other minimum cover by a finite sequence of level transformations (Theorem 2). Thus we have a procedure for finding all minimum covers: First, reduce any cover to a minimum cover by the algorithm; second, apply the level transformations to obtain all other minimum covers. The treatment of the maximum matching theorem and its corresponding algorithm, in the following section, closely parallels that given for minimum covers. The final section establishes a relationship between minimum covers and maximum matchings. Both the minimum cover theorem and the maximum matching theorem were first proved by induction. The inductive proof of the maximum matching theorem, together with a discussion of its relevance to two similar theorems dealing with point-covers and pointmatchings, are given in Berge [l]. The problem of finding a simple algorithm for obtaining a minimum cover of a graph was proposed by Paul Roth as the one-dimensional instance of the more general question of finding a minimum cover for a cubical complex. This problem is a topological formulation of the synthesis of a switching system with minimum cost (cf. Roth [3]).