A Symmetric Strategy in Graph Avoidance Games

Summary

ABSTRACT. In the graph avoidance game two players alternately color the edges of a graph G in red and in blue respectively. The player who first creates a monochromatic subgraph isomorphic to a forbidden graph F loses. A symmetric strategy of the second player ensures that, independently of the first player's strategy, the blue and the red subgraph are isomorphic after every round of the game. We address the class of those graphs G that admit a symmetric strategy for all F and discuss relevant graph-theoretic and complexity issues. We also show examples when, though a symmetric strategy on G generally does not exist, it is still available for a particular F.

1. Introduction

In a broad class of games that have been studied in the literature, two players, A and *B, alternately color the edges of a graph G in red and in blue respectively. In the achievement game the objective is to create a monochromatic subgraph isomorphic to a given graph F. In the avoidance game the objective is, on the contrary, to avoid creating such a subgraph. Both the achievement and the avoidance games have strong and weak versions. In the strong version A and 23 both have the same objective. In the weak version B just plays against A, that is, tries either to prevent A from creating a copy of F in the achievement game or to force such creation in the avoidance game. The weak achievement game, known also as the Maker-Breaker game, is most studied [4; 1; 13]. Our paper is motivated by the strong avoidance game [7; 5] where monochromatic F-subgraphs of G are forbidden, and the player who first creates such a subgraph loses.

The instance of a strong avoidance game with complete graphs G = K₆ and F = K₃ is well known under the name SIM [15]. Since for any edge bicoloring of K₆ there is a monochromatic K₃ , a draw in this case is impossible. It is proven in [12] that a winning strategy in SIM is available for B. A few other results for small graphs are known [7].