A Symmetric Strategy in Graph Avoidance Games

In the graph avoidance game two players alternatingly color 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 . 1Computer Science Department, New Mexico State University, Las Cruces, NM 88003, USA. 2Institut für Informationssysteme, Technische Universität Wien, Favoritenstraße 9, A-1040 Wien, Austria. 3Research partly supported by Austrian Science Foundation grant Z29-INF. 4Department of Mechanics & Mathematics, Lviv University, Universytetska 1, 79000 Lviv, Ukraine. 5Research was partly done while visiting the Institut für Informationssysteme at the Technische Universität Wien, supported by a Lise Meitner Fellowship of the Austrian Science Foundation (FWF grant M 532). A preliminary version of this paper appeared in the Proceedings of MSRI-CGTR’2000, the 2nd Combinatorial Game Theory Research Workshop at the Mathematical Sciences Research Institute in Berkeley, California. Online version of this paper: http://www.dbai.tuwien.ac.at/staff/slany/pubs/dbai-tr-2001-42.ps.gz Copyright c © 2008 Frank Harary, Wolfgang Slany, and Oleg Verbitsky TECHNICAL REPORT DBAI-TR-2001-42 2