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

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