In the Avoider-Enforcer game on the complete graph $K_n$, the players(Avoider and Enforcer) each take an edge in turn. Given a graph property$\mathcal{P}$, Enforcer wins the game if Avoider's graph has the property$\mathcal{P}$. An important parameter is $\tau_E({\cal P})$, the smallestinteger $t$ such that Enforcer can win the game against any opponent in $t$rounds. In this paper, let $\mathcal{F}$ be an arbitrary family of graphs and$\mathcal{P}$ be the property that a member of $\mathcal{F}$ is a subgraph oris an induced subgraph. We determine the asymptotic value of$\tau_E(\mathcal{P})$ when $\mathcal{F}$ contains no bipartite graph andestablish that $\tau_E(\mathcal{P})=o(n^2)$ if $\mathcal{F}$ contains abipartite graph. The proof uses the game of JumbleG and the Szemer\'edi Regularity Lemma.
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