Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-$\eta$ Deletion
For a constant $\eta \geq 0$, Pathwidth-$\eta$ Deletion is the problem of deciding whether, for a given graph $G$ and integer $k$, there is a set $S \subseteq V(G)$ of size at most $k$ such that the pathwidth of $G - S$ is at most $\eta$. The problems Treewidth-$\eta$ Deletion and Treedepth-$\eta$ Deletion are defined similarly for the parameters treewidth and treedepth, respectively. A landmark result of Fomin et al. [FOCS, 2012] shows that, for any constant $\eta$, all three problems admit a kernel on $O(k^{c(\eta)})$ vertices, where $c(\eta)$ is a constant depending on $\eta$. Giannopoulou et al. [ACM TALG, 2017] show that, in some sense, this result is optimal for Treewidth-$\eta$ Deletion: for $\eta \geq 2$ and even when parameterizing by the size of a vertex cover $M$ of the input graph, there is no kernel of size $O(|M|^{\frac{\eta+1}{2}-\varepsilon})$, for any $\varepsilon > 0$. Contrasting this result, they prove that Treedepth-$\eta$ Deletion admits a uniform polynomial kernel, that is, a kernel of size $O(k^c)$ for a constant $c$ that is independent of $\eta$. In comparison, the question whether Pathwidth-$\eta$ Deletion admits a uniform polynomial kernel has been neglected in the literature. As treewidth and pathwidth tend to behave similarly, it is natural to expect that no uniform kernel exists when parameterizing by the size of a vertex cover. Surprisingly, we show this not to be the case. More concretely, we prove the existence of a uniform polynomial kernel for Pathwidth-$\eta$ Deletion when parameterizing by (1) the solution size $k$ plus the size of a set $M$ such that $G - M$ has bounded treedepth; (2) the (vertex-deletion) distance to pathwidth-$1$ graphs; (3) the distance to the class of graphs with treedepth at most $\eta + 1$. This leads us to conjecture that Pathwidth-$\eta$ Deletion admits a uniform kernel when parameterizing by the solution size $k$.