A Dividing Line for Structural Kernelization of Component Order Connectivity via Distance to Bounded Pathwidth

In this work we study a classic generalization of the Vertex Cover (VC) problem, called the Component Order Connectivity (COC) problem. In COC, given an undirected graph $G$, integers $d \geq 1$ and $k$, the goal is to determine if there is a set of at most $k$ vertices whose deletion results in a graph where each connected component has at most $d$ vertices. When $d=1$, this is exactly VC. This work is inspired by polynomial kernelization results with respect to structural parameters for VC. On one hand, Jansen & Bodlaender [TOCS 2013] show that VC admits a polynomial kernel when the parameter is the distance to treewidth-$1$ graphs, on the other hand Cygan, Lokshtanov, Pilipczuk, Pilipczuk & Saurabh [TOCS 2014] showed that VC does not admit a polynomial kernel when the parameter is distance to treewidth-$2$ graphs. Greilhuber & Sharma [IPEC 2024] showed that, for any $d \geq 2$, $d$-COC cannot admit a polynomial kernel when the parameter is distance to a forest of pathwidth $2$. Here, $d$-COC is the same as COC only that $d$ is a fixed constant not part of the input. We complement this result and show that like for the VC problem where distance to treewidth-$1$ graphs versus distance to treewidth-$2$ graphs is the dividing line between structural parameterizations that allow and respectively disallow polynomial kernelization, for COC this dividing line happens between distance to pathwidth-$1$ graphs and distance to pathwidth-$2$ graphs. The main technical result of this work is that COC admits a polynomial kernel parameterized by distance to pathwidth-$1$ graphs plus $d$.

💡 Research Summary

The paper investigates the Component Order Connectivity (COC) problem, a natural generalization of Vertex Cover (VC). An instance consists of an undirected graph G, integers d ≥ 1 and k; the task is to decide whether there exists a set S of at most k vertices whose removal leaves a graph in which every connected component contains at most d vertices. When d = 1 the problem coincides with VC, while for larger d it models a broader class of network‑vulnerability measures.

Motivated by the rich kernelization landscape for VC, the authors focus on structural parameterizations rather than the solution size k. Prior work showed that VC admits a polynomial kernel when parameterized by the distance to treewidth‑1 graphs (equivalently, the feedback‑vertex‑set number) but not when parameterized by the distance to treewidth‑2 graphs. Greilhuber and Sharma (IPEC 2024) extended the negative side to COC, proving that for any fixed d ≥ 2, d‑COC has no polynomial kernel when the parameter is the distance to a forest of pathwidth 2.

The central contribution of this work is to identify the exact “dividing line” for COC: the distance to pathwidth‑1 graphs (plus d) yields a polynomial kernel, whereas the distance to pathwidth‑2 graphs does not. Formally, the authors define two parameterized problems:

1. COC / d + pw‑1 – Input: graph G, integer d, and a modulator M such that G − M has pathwidth ≤ 1. Parameter: |M| + d.
2. d‑COC / pw‑1 – Same as above but d is a fixed constant; parameter: |M|.

The main theorem states that COC / d + pw‑1 admits a kernel with O(d⁷|M|³ + d⁶|M|⁴) vertices, and for each fixed d, d‑COC / pw‑1 admits a kernel with O(|M|⁴) vertices. This positively answers an open question of Greilhuber & Sharma regarding kernels for distance‑to‑linear‑forest parameterizations.

Technically, the result cannot rely on the treedepth‑based approach of Jansen and Pieterse (which works for distance to treedepth‑η modulators) because graphs of pathwidth 1 may contain arbitrarily long paths and thus have unbounded treedepth. Instead, the authors develop new reduction rules that directly bound the size of the connected components after removing the modulator. A key tool is the notion of d‑blocking sets: a set X of vertices such that no minimum d‑coc set contains any vertex of X. By iteratively identifying minimal d‑blocking sets and moving them into the modulator, they ensure that the number of components in G − M shrinks to |M|·O(1). This process builds on a lifted version of a component‑reduction technique originally devised for VC by Hols, Kratsch, and Pieterse.

Beyond the primary dichotomy, the paper explores several related directions:

• F‑MinorDeletion perspective: COC can be expressed as an F‑MinorDeletion problem (e.g., d‑COC corresponds to deleting a P_{d+1} minor). Existing kernelization dichotomies for F‑MinorDeletion rely on parameters such as bounded bridgedepth or bounded elimination distance to the F‑minor‑free class. The authors show that d‑COC falls outside these known regimes: it admits a polynomial kernel for distance to pathwidth‑1 graphs, a class that does not have bounded elimination distance to the relevant minor‑free graphs. Thus d‑COC provides the first known F‑MinorDeletion problem whose kernelization boundary is governed by a new structural parameter.

• Maximum‑degree modulators: Using the main kernel, they obtain a polynomial kernel for COC / d + dist‑to‑G when G is the class of graphs of maximum degree 2. Conversely, they prove that for planar graphs of maximum degree 3, d‑COC / dist‑to‑G admits no kernel of any size unless P = NP, establishing a sharp dichotomy with respect to degree‑bounded modulators.

• General component‑reduction framework: The authors present a generic preprocessing routine (Theorem 4.13) that, for any graph class G closed under disjoint union and admitting a polynomial kernel for d‑COC / dist‑to‑G, reduces the number of components of G − M to |M|·O(1). This routine hinges on the existence of bounded minimal d‑blocking sets in G, a property that holds for many natural classes (e.g., caterpillars, bounded‑degree forests).

• Hardness results: They show that COC parameterized by vertex cover number plus solution size k does not admit a polynomial kernel unless NP ⊆ coNP/poly, and that COC parameterized solely by distance to pathwidth‑1 is W