A characterization of $K_{2,4}$-minor-free graphs

TL;DR

This paper provides a complete structural characterization of graphs that do not contain a $K_{2,4}$ minor, describing their structure based on connectivity and planarity properties, including specific graph families and decompositions.

Contribution

It introduces a detailed structural description of $K_{2,4}$-minor-free graphs, including classifications for 2- and 3-connected cases and new characterizations involving $xy$-outerplanar graphs.

Findings

3-connected $K_{2,4}$-minor-free graphs are either small or planar with specific properties.

2-connected $K_{2,4}$-minor-free graphs are unions of $xy$-outerplanar graphs or derived from 3-connected graphs.

$xy$-outerplanar graphs are characterized by the absence of rooted $K_{2,2}$-minors.

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 $K_4$ and, for each $n \ge 5$, $2n-8$ nonisomorphic graphs of order $n$. 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.