On terminal delta-wye reducibility of planar graphs

A graph is terminal ∆ − Y -reducible if, it can be reduced to a distinguished set of terminal vertices by a sequence of series-parallel reductions and ∆−Y -transformations. Terminal vertices (o terminals for short) cannot be deleted by reductions and transformations. Reducibility of terminal graphs is very difficult and in general not possible for graphs with more than three terminals (even planar graphs). Terminal reducibility plays an important role in decomposition theorems in graph theory and in important applications, as for example, network reliability. We prove terminal reducibility of planar graphs with at most three terminals. The most important consequence of our proof is that this implicitly gives an efficient algorithm, of order O(n4), for reducibility of planar graphs with at most three terminals that also can be used for restricted reducibility problems with more terminals. It is well known that these operations can be translated to operations on the medial graph. Our proof makes use of this translation in a novel way, furthermore terminal vertices now seen as terminal faces and by duality of the reductions and transformations, the set of terminals can be taken as a set of vertices and a set of faces of the original graph. Corresponding author. Partially supported by CONACyT grants U40201, 45256 and SNI. Partially supported by CONACyT grants U40201, 45256 and SNI. 1