Branching programs are a well-established computation
model for boolean functions, especially read-once
branching programs (BP1s) have been studied intensively.
A very simple function f in n^2 variables is
exhibited such that both the function f and its negation
\neg f can be computed by \Sigma^3_p-circuits,
the function f has nondeterministic BP1s
(with one nondeterministic node) of linear size and \neg f
has size O(n^4) for oblivious nondeterministic BP1s
but f requires nondeterministic graph-driven BP1s
of size 2^{\Omega(n)}.
This answers an open question stated by
Jukna, Razborov, Savick\'y, and Wegener (1999).