  Given a fixed graph $H$ with at least two edges and positive integers $n$ and
$b$, the strict $(1 \colon b)$ Avoider-Enforcer $H$-game, played on the edge
set of $K_n$, has the following rules: In each turn Avoider picks exactly one
edge, and then Enforcer picks exactly $b$ edges. Avoider wins if and only if
the subgraph containing her/his edges is $H$-free after all edges of $K_n$ are
taken.
  The lower threshold of a graph $H$ with respect to $n$ is the largest $b_0$
for which Enforcer has a winning strategy for the $(1\colon b)$ $H$-game played
on $K_n$ for any $b \leq b_0$, and the upper threshold is the largest $b$ for
which Enforcer wins the $(1 \colon b)$ game. The separation conjecture of
Hefetz, Krivelevich, Stojakovi\'c and Szab\'o states that for any connected
$H$, the lower threshold and the upper threshold of the Avoider-Enforcer
$H$-game played on $K_n$ are not of the same order in $n$. Until now, the
conjecture has been verified only for stars, by Grzesik, Mikala\v{c}ki, Nagy,
Naor, Patkos and Skerman.
  We show that the conjecture holds for every connected graph $H$ with at most
one cycle (and at least two edges), with a polynomial separation between the
lower and upper thresholds. We also prove an upper bound for the lower
threshold of any graph $H$ with at least two edges, and show that this bound is
tight for all graphs in which each connected component contains at most one
cycle. Along the way, we establish number-theoretic tools that might be useful
for other problems of this type.
