Rooted topological minors on four vertices
Koyo Hayashi, Ken‐ichi Kawarabayashi · Journal of Combinatorial Theory Series B · 2021
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={v1,v2,v3,v4}, there are three internally disjoint paths P1,P2,P3 with end vertices v1,v2 with v3,v4 on P1,P2, respectively. Therefore, this yields a K4−-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.