  For an $n$-vertex graph $G$, let $h(G)$ denote the smallest size of a subset
of $V(G)$ such that it intersects every maximum independent set of $G$. A
conjecture posed by Bollob\'{a}s, Erd\H{o}s and Tuza in early 90s remains
widely open, asserting that for any $n$-vertex graph $G$, if the independence
number $\alpha(G) =\Omega(n) $, then $h(G) = o(n)$. In this paper, we establish
the validity of this conjecture for various classes of graphs, including disk
graphs, even-hole-free graphs, circle graphs, and those hereditary graphs
having sublinear balanced separators. We also determine the exact values of
smallest possible hitting sets in comparability graphs, incomparability graphs
and the graphs with VC-dimension one.
