Rooted K4-Minors

Let a,b,c,d be four vertices in a graph G. A K-4-minor rooted at a,b,c,d consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of G, respectively containing a,b,c,d. We characterise precisely when G contains a K-4-minor rooted at a,b,c,d by describing six classes of obstructions, which are the edge-maximal graphs containing no K-4-minor rooted at a,b,c,d. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a K-4-minor rooted at a,b,c,d for every choice of a,b,c,d. (2) A 3-connected planar graph contains a K-4-minor rooted at a,b,c,d if and only if a,b,c,d are not on a single face.