Hitting all maximum independent sets

We describe an infinite family of graphs $G_n$, where $G_n$ has $n$ vertices, independence number at least $n\mathrm{/}4$, and no set of less than $\sqrt{n}\mathrm{/}2$ vertices intersects all its maximum independent sets. This is motivated by a question of Bollob'as, Erd\H{o}s and Tuza, and disproves a recent conjecture of Friedgut, Kalai and Kindler. Motivated by a related question of the last authors, we show that for every graph $G$ on $n$ vertices with independence number $(1\mathrm{/}4+\text{eps})n$, the average independence number of an induced subgraph of $G$ on a uniform random subset of the vertices is at most $(1\mathrm{/}4+\text{eps}-\mathrm{\Omega }({\text{eps}}^2))n$.