On the delta‐wye reduction for planar graphs

Abstract

We provide an elementary proof of an important theorem by G. V. Epifanov, according to which every two-terminal planar graph satisfying certain connectivity restrictions can by some sequence of series/parallel reductions and delta-wye exchanges be reduced to the graph consisting of the two terminals and just one edge.

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

“…Truemper [21] mentioned that any 2-connected minor of a 2-connected Y ∆Y -reducible graph was itself Y ∆Y -reducible. Gitler [13] extended this result to graphs with terminals.…”

“…There are several other reduction algorithms given in 2.4 and [21]. Neither of these were concerned with terminals.…”

“…This was first proved by Epifanov [9] in 1966 (see also [15]). Simpler proofs are given by Feo and Provan [11], and Truemper [21]. Gitler [13] showed that graphs with no K 5 minor are Y ∆Y -reducible, and also that graphs with no K 3,3 minor are Y ∆Y -reducible.…”

“…Truemper [21] showed that the class of Y ∆Y -reducible graphs is minor closed (he assumed that the minor was 2-connected, since he did not allow degree-one reductions). From the above, it follows that there are a finite number of minorminimal graphs that are not Y ∆Y -reducible.…”

Tightening Curves on Surfaces via Local Moves

“…Specifically, Steinitz proved that any non-simple closed curve with no empty monogons contains a minimal bigon which can be emptied with 3 3 moves and then removed by a 2 0 move. This quadratic bound has been reproved and generalized by several other authors [23,24,30,40,48,49]. Chang and Erickson recently improved the upper bound to O(n 3/2 ) and proved that such bound is optimal in the worst case [11].…”