A Halin graph is a plane graph constructed from a planar drawing of a tree by connecting all leaves of the tree with a cycle which passes around the boundary of the graph. The tree must have four or more vertices and no vertices of degree two. Halin graphs have many nice properties such as being Hamiltonian and remaining Hamiltonian after any single vertex deletion. In 1975, Lovász and Plummer conjectured that every 4-connected plane triangulation contains a spanning Halin subgraph. We recently gave a negative answer to this conjecture. In this paper, we construct an infinite class of 5-connected plane triangulations without a spanning Halin subgraph. Our smallest example contains 512 vertices.

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 $\{x_1 y_1, x_2 y_2, \ldots, x_k y_k\}$ satisfying a certain condition by an $x_i y_i$-outerplanar graph. From our characterization it follows that a $K_{2,4}$-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 $K_{2,4}$-minor-free graph is either planar or else toroidal and projective-planar.

A \sl Halin graph is a simple plane graph consisting of a tree without degree 2 vertices and a cycle induced by the leaves of the tree. In 1975, Lovász and Plummer conjectured that every 4-connected plane triangulation has a spanning Halin subgraph. In this paper, we construct an infinite family of counterexamples to the conjecture.