  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.
