Simpler analysis of LP extreme points for traveling salesman and survivable network design problems

We consider the Survivable Network Design Problem (SNDP) and the Symmetric Traveling Salesman Problem (STSP). We give simpler proofs of the existence of a 12-edge and 1-edge in any extreme point of the natural LP relaxations for the SNDP and STSP, respectively. We formulate a common generalization of both problems and show our results by a new counting argument. We also obtain a simpler proof of the existence of a 12-edge in any extreme point of the set-pair LP relaxation for the element connectivitySurvivable Network Design Problem (SNDPelt).