An optimal solution of the Shannon switching game played on a graph

Abstract

In a previous paper of the writer's, co-authored with John Bruno, a simple graph-theoretic solution to the Shannon two-person switching game was presented. The solution is constructive in that efficient algorithms were formulated that determined if a game played on any given graph is a short, cut, or neutral game, and gave the strategies for winning each type of game. The proof makes use of Lehman's solution combined with the principal partition of a graph, a unique decomposition of a graph discovered by Kishi and Kajitani. In addition, by a correct application of duality theory Bruno and Weinberg also proved that the strategies are global strategies, that is, they guarantee a win in the so-called global game for the short, cut, and neutral games. Another of their proofs showed that a further decomposition of the principal partition of Kishi and Kajitani was possible, that is, a refined unique partition can be obtained. Finally, they generalized all the new results on the principal partition to matroids so that the complete analysis can be automatically applied to solve the Shannon game played on a matroid. In this paper we improve the solution to an optimal one. This optimal solution also generalizes to the game played on a matroid, but here we focus attention on the graph-theoretic case and present the solution entirely within the context of graph theory. We accomplish this by using an enhanced version of graph theory that we claim is a new paradigm for graph theory (which we present in a companion paper). By an optimal solution we mean that a global game is won in a minimum number of plays. This is brought about by determining at the beginning of the game the smallest possible minor of the given graph on which a winning game with respect to the distinguished edge can be played. The size of this minor is automatically given by the co-spanning sets of the minor formed by the union of those atoms, and only those atoms, of the associated partial order, that are required for guaranteeing the winning game. The latter minor is thus unique, and the minimum number of plays for the global game is given by the strategies and determined by the size of the co-spanning sets in the smallest possible minor. Just as the previous solution was based on the principal partition, so the optimal solution is based on a refinement of the principal partition, which we designate as the general principal partition, whose development was begun by Bruno and Weinberg using concepts and algorithms first formulated by them and then applied in a further and final extension independently by Narayanan and Tomizawa. The optimal solution uses the same strategies and efficient algorithms (but now applied to the smallest minor) as the earlier solution and thus continues to be constructive. Finally, there are also versions of the Shannon switching game played on a directed graph and an oriented matroid, at least two of which have a solution with the same necessary and sufficient conditions as the corresponding undirected games of the previous paper. Hence the optimal solution also applies to them. Separate papers will present the optimal solutions to the Shannon game played on a matroid, on a directed graph and on an oriented matroid.