T-joins intersecting small edge-cuts in graphs

Abstract

In an earlier paper 3, we studied cycles in graphs that intersect all edge-cuts of prescribed sizes. Passing to a more general setting, we examine the existence of T-joins in grafts that intersect all edge-cuts whose size is in a given set A ⊆{1,2,3}. In particular, we characterize all the contraction-minimal grafts admitting no T-joins that intersect all edge-cuts of size 1 and 2. We also show that every 3-edge-connected graft admits a T-join intersecting all 3-edge-cuts. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 64–71, 2007