Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size

In the $\mathcal{F}$-Minor-Free Deletion problem one is given an undirected graph $G$, an integer $k$, and the task is to determine whether there exists a vertex set $S$ of size at most $k$, so that $G-S$ contains no graph from the finite family $\mathcal{F}$ as a minor. It is known that whenever $\mathcal{F}$ contains at least one planar graph, then $\mathcal{F}$-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size $k^{\mathcal{O}(1)}$ [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size. We study the Outerplanar Deletion problem, in which we want to remove at most $k$ vertices from a graph to make it outerplanar. This is a special case of $\mathcal{F}$-Minor-Free Deletion for the family $\mathcal{F} = \{K_4, K_{2,3}\}$. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with $\mathcal{O}(k^4)$ vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size $k$ has $\mathcal{O}(k^4)$ vertices and edges.