Characterizing indecomposable plane continua from their complements
  We show that a plane continuum X is indecomposable iff X has a sequence (U_n) of not necessarily distinct complementary domains satisfying what we call the double-pass condition: If one draws an open arc A_n in each U_n whose ends limit into the boundary of U_n, one can choose components of U_n minus A_n whose boundaries intersected with the continuum (which we call shadows) converge to the continuum. 
Clinton P. Curry, John C. Mayer, E. D. Tymchatyn