Four-terminal reducibility and projective-planar wye-delta-wye-reducible graphs

A graph is Y Y-reducible if it can be reduced to a vertex by a sequence of series-parallel reductions and Y Y-transformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that four-terminal planar graphs are Y Y-reducible when at least three of the vertices lie on the same face. Using this result we characterize Y Y-reducible projective-planar graphs. We also consider terminals in projective-planar graphs, and establish that graphs of crossing-number one are Y Y-reducible.