  We prove two theorems which allow one to recognize indecomposable subcontinua
of closed surfaces without boundary. If $X$ is a subcontinuum of a closed
surface $S$, we call the components of $S \setminus X$ the complementary
domains of $X$. We prove that a continuum is either indecomposable or the union
of two indecomposable continua whenever it has a sequence of distinct
complementary domains whose boundaries limit to the continuum in the Hausdorff
metric. We define a slightly stronger condition on the complementary domains of
a continuum, called the double-pass condition, which we conjecture is
equivalent to indecomposability of the continuum. We prove that this is so for
continua which are not the boundary of one of their complementary domains.
