  A widely open conjecture proposed by Bollob\'as, Erd\H{o}s, and Tuza in the
early 1990s states that for any $n$-vertex graph $G$, if the independence
number $\alpha(G) = \Omega(n)$, then there is a subset $T \subseteq V(G)$ with
$|T| = o(n)$ such that $T$ intersects all maximum independent sets of $G$. In
this paper, we prove that this conjecture holds for graphs that do not contain
an induced $K_{s,t}$ for fixed $t \ge s$. Our proof leverages the probabilistic
method at an appropriate juncture.
