  We study the complexity of the following cell connection and separation
problems in segment arrangements. Given a set of straight-line segments in the
plane and two points $a$ and $b$ in different cells of the induced arrangement:
  (i) compute the minimum number of segments one needs to remove so that there
is a path connecting $a$ to $b$ that does not intersect any of the remaining
segments; (ii) compute the minimum number of segments one needs to remove so
that the arrangement induced by the remaining segments has a single cell; (iii)
compute the minimum number of segments one needs to retain so that any path
connecting $a$ to $b$ intersects some of the retained segments.
  We show that problems (i) and (ii) are NP-hard and discuss some special,
tractable cases. Most notably, we provide a linear-time algorithm for a variant
of problem (i) where the path connecting $a$ to $b$ must stay inside a given
polygon $P$ with a constant number of holes, the segments are contained in $P$,
and the endpoints of the segments are on the boundary of $P$. For problem (iii)
we provide a cubic-time algorithm.
