  The complement of a complex hyperplane arrangement is known to be homotopic
to a minimal CW complex. There are several approaches to the minimality. In
this paper, we restrict our attention to real two dimensional cases, and
introduce the "dual" objects so called minimal stratifications. The strata are
explicitly described as semialgebraic sets. The stratification induces a
partition of the complement into a disjoint union of contractible spaces, which
is minimal in the sense that the number of codimension $k$ pieces equals the
$k$-th Betti number.
  We also discuss presentations for the fundamental group associated to the
minimal stratification. In particular, we show that the fundamental groups of
complements of a real arrangements have positive homogeneous presentations.
