A Characterization of $K_{2,4}$-Minor-Free Graphs

Abstract

We provide a complete structural characterization of $K_{2,4}$-minor-free graphs. The 3-connected $K_{2,4}$-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 $n \geq 5$ as well as $K_4$. To describe the 2-connected $K_{2,4}$-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 $K_{2,2}$ minor where $x$ and $y$ correspond to the two vertices on one side of the bipartition of $K_{2,2}$. Each 2-connected $K_{2,4}$-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 $K_{2,4}$-minor-free graph by replacing each edge $x_iy_i$ in a set $\{x_1 y_1, x...