Preprocessing Subgraph and Minor Problems: When Does a Small Vertex Cover Help?

We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like q-Coloring, Odd Cycle Transversal, Chordal Deletion, Eta Transversal, or Long Path, parameterized by the size of a vertex cover, but also imply new polynomial kernels for problems like F-Minor-Free Deletion, which is to delete at most k vertices to obtain a graph with no minor from a fixed finite set F. While our characterization captures many interesting problems, the kernelization complexity landscape of parameterizations by vertex cover is much more involved. We demonstrate this by several results about induced subgraph and minor containment testing, which we find surprising. While it was known that testing for an induced complete subgraph has no polynomial kernel unless NP is in coNP/poly, we show that the problem of testing if a graph contains a complete graph on t vertices as a minor admits a polynomial kernel. On the other hand, it was known that testing for a path on t vertices as a minor admits a polynomial kernel, but we show that testing for containment of an induced path on t vertices is unlikely to admit a polynomial kernel.

💡 Research Summary

The paper investigates kernelization for graph problems when the parameter is the size of a given vertex cover (vc) of the input graph. It establishes three broad, easily verifiable conditions that guarantee the existence of polynomial‑size kernels. The first condition requires that the solution depends only on the vertices inside the cover or that interactions between the cover and its complement are limited. The second condition demands that the subgraph induced by the cover can be safely reduced—by replacing or deleting certain bounded‑size patterns—without affecting the existence of a solution. The third condition asserts that the essential structure needed to solve the problem is a bounded‑size pattern that must appear inside the cover, and its size is polynomial in vc. When all three hold, the instance can be compressed to a size polynomial in vc.

Using this framework, the authors re‑explain known polynomial kernels for a variety of problems: q‑Coloring, Odd Cycle Transversal, Chordal Deletion, η‑Transversal, and Long Path, all parameterized by vc. In each case the problem either reduces to a simple assignment on the cover vertices or to checking a limited set of configurations inside the cover, satisfying the three conditions.

The paper also derives new polynomial kernels. The most notable is for F‑Minor‑Free Deletion, where the goal is to delete at most k vertices to obtain a graph that excludes any minor from a fixed finite family F. By observing that any forbidden minor must be “realized” inside the vertex cover, the authors design reduction rules that shrink the instance to O(vc^c) vertices for some constant c, thereby providing the first polynomial kernel for this problem under the vc‑parameter.

Beyond positive results, the authors explore the limits of kernelization under the same parameter. They show a striking dichotomy between minor containment and induced subgraph containment. Testing whether a graph contains a complete graph K_t as a minor admits a polynomial kernel: the existence of a K_t‑minor can be certified by a set of t vertices in the cover that can be linked via edge‑contractions, a structure that can be bounded and reduced. In contrast, testing for a K_t induced subgraph (a clique) is known not to admit a polynomial kernel unless NP ⊆ coNP/poly, because any kernel would imply a compression of the CLIQUE problem beyond known limits.

Similarly, while the t‑vertex path as a minor problem already has a polynomial kernel, the authors prove that testing for an induced t‑vertex path is unlikely to have one under standard complexity assumptions. The induced path problem requires checking all possible vertex orderings, which cannot be captured by a bounded pattern inside the vertex cover, violating the third condition.

These results illustrate that the vertex‑cover parameter behaves very differently from other structural parameters such as treewidth or pathwidth. Because a vertex cover directly isolates a small “core” of the graph, many otherwise hard problems become amenable to aggressive reduction, yet some problems remain resistant due to the nature of induced subgraph requirements. The paper thus provides both a unifying theory for existing kernels and a roadmap for identifying new kernelizable problems, while also delineating clear boundaries where kernelization is unlikely to succeed.