  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 smallest
integer $t$ 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 or
is an induced subgraph. We determine the asymptotic value of
$\tau_E(\mathcal{P})$ when $\mathcal{F}$ contains no bipartite graph and
establish that $\tau_E(\mathcal{P})=o(n^2)$ if $\mathcal{F}$ contains a
bipartite graph.
  The proof uses the game of JumbleG and the Szemer\'edi Regularity Lemma.
