A Characterization of K2, 4-Minor-Free Graphs

We provide a complete structural characterization of K2,4-minor-free graphs. The 3-connected K2,4minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains K4 and, for each n ≥ 5, 2n − 8 nonisomorphic graphs of order n. To describe the 2-connected K2,4-minor-free graphs we use xy-outerplanar graphs, graphs embeddable in the plane with a Hamilton xy-path so that all other edges lie on one side of this path. We show that, subject to an appropriate connectivity condition, xy-outerplanar graphs are precisely the graphs that have no rooted K2,2-minor where x and y correspond to the two vertices on one side of the bipartition of K2,2. Each 2-connected K2,4-minor-free graph is then (i) outerplanar, (ii) the union of three xy-outerplanar graphs and possibly the edge xy, or (iii) obtained from a 3-connected K2,4-minor-free graph by replacing each edge xiyi in a set {x1y1, x2y2, . . . , xkyk} satisfying a certain condition by an xiyi-outerplanar graph.