  Given a graph $G$, we consider a game where two players, $A$ and $B$,
alternatingly color edges of $G$ in red and in blue respectively. Let $l(G)$ be
the maximum number of moves in which $B$ is able to keep the red and the blue
subgraphs isomorphic, if $A$ plays optimally to destroy the isomorphism. This
value is a lower bound for the duration of any avoidance game on $G$ under the
assumption that $B$ plays optimally. We prove that if $G$ is a path or a cycle
of odd length $n$, then $\Omega(\log n)\le l(G)\le O(\log^2 n)$. The lower
bound is based on relations with Ehrenfeucht games from model theory. We also
consider complete graphs and prove that $l(K_n)=O(1)$.
