T-Joins intersecting small edge-cuts in graphs

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 subset of {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. (C) 2007 Wiley Periodicals.