  In this paper, we study a conjecture of Andries E. Brouwer from 1996
regarding the minimum number of vertices of a strongly regular graph whose
removal disconnects the graph into non-singleton components.
  We show that strongly regular graphs constructed from copolar spaces and from
the more general spaces called $\Delta$-spaces are counterexamples to Brouwer's
Conjecture. Using J.I. Hall's characterization of finite reduced copolar
spaces, we find that the triangular graphs $T(m)$, the symplectic graphs
$Sp(2r,q)$ over the field $\mathbb{F}_q$ (for any $q$ prime power), and the
strongly regular graphs constructed from the hyperbolic quadrics $O^{+}(2r,2)$
and from the elliptic quadrics $O^{-}(2r,2)$ over the field $\mathbb{F}_2$,
respectively, are counterexamples to Brouwer's Conjecture. For each of these
graphs, we determine precisely the minimum number of vertices whose removal
disconnects the graph into non-singleton components. While we are not aware of
an analogue of Hall's characterization theorem for $\Delta$-spaces, we show
that complements of the point graphs of certain finite generalized quadrangles
are point graphs of $\Delta$-spaces and thus, yield other counterexamples to
Brouwer's Conjecture.
  We prove that Brouwer's Conjecture is true for many families of strongly
regular graphs including the conference graphs, the generalized quadrangles
$GQ(q,q)$ graphs, the lattice graphs, the Latin square graphs, the strongly
regular graphs with smallest eigenvalue -2 (except the triangular graphs) and
the primitive strongly regular graphs with at most 30 vertices except for few
cases.
  We leave as an open problem determining the best general lower bound for the
minimum size of a disconnecting set of vertices of a strongly regular graph,
whose removal disconnects the graph into non-singleton components.
