A CHARACTERIZATION OF K-2,K-4-MINOR-FREE GRAPHS

We provide a complete structural characterization of K-2,K-4-minor-free graphs. The 3-connected K-2,K-4-minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains 2n-8 nonisomorphic graphs of order n for each n >= 5 as well as K-4. To describe the 2-connected K-2,K-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 K-2,K-2 minor where x and y correspond to the two vertices on one side of the bipartition of K-2,K-2. Each 2-connected K-2,K-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 K-2,K-4-minor-free graph by replacing each edge in a set {x(1)y(1), x(2)Y(2), . . . , XkYk} satisfying a certain condition by an x(i)y(i)-outerplanar graph. From our characterization it follows that a K-2,K-4-minor-free graph has a Hamilton cycle if it is 3-connected and a Hamilton path if it is 2-connected. Also, every 2-connected K-2,K-4-minor-free graph is either planar or else toroidal and projective-planar.