Rooted topological minors on four vertices

For a graph G and a set Z of four distinct vertices of G, a diamond on Z is a subgraph of G such that, for some labeling Z = { v 1 , v 2 , v 3 , v 4 } , there are three internally disjoint paths P 1 , P 2 , P 3 with end vertices v 1 , v 2 with v 3 , v 4 on P 1 , P 2 , respectively. Therefore, this yields a K 4 − -subdivision with branch vertices on Z. We characterize graphs G that contain no diamond on a prescribed set Z of four vertices, under the assumption that for every v ∈ Z there are three paths of G from v to Z − { v } , mutually disjoint except for v. Moreover, we can find two “different” such subdivisions, if one exists. Our proof is based on Mader's S-paths theorem.