Towards optimal kernel for connected vertex cover in planar graphs

We study the parameterized complexity of the connected version of the vertex cover problem, where the solution set has to induce a connected subgraph. Although this problem does not admit a polynomial kernel for general graphs (unless NP is a subset of coNP/poly), for planar graphs Guo and Niedermeier [ICALP'08] showed a kernel with at most 14k vertices, subsequently improved by Wang et al. [MFCS'11] to 4k. The constant 4 here is so small that a natural question arises: could it be already an optimal value for this problem? In this paper we answer this quesion in negative: we show a (11/3)k-vertex kernel for Connected Vertex Cover in planar graphs. We believe that this result will motivate further study in search for an optimal kernel.

💡 Research Summary

The paper investigates the parameterized complexity of the Connected Vertex Cover (CVC) problem when the input graph is planar. While the problem is known to be NP‑complete even on planar graphs and does not admit a polynomial kernel on general graphs (unless NP ⊆ coNP/poly), previous work by Guo and Niedermeier gave a 14k‑vertex kernel for planar instances, later improved to a 4k‑vertex kernel by Wang et al. The authors ask whether the constant 4 is optimal. They answer this negatively by presenting a kernel with at most (11/3) k vertices, thereby lowering the linear factor from 4 to approximately 3.667.

The kernelization algorithm proceeds in three phases.

Phase 1 – Basic reduction rules.
The algorithm first applies a set of reduction rules. Rules 1‑3 are exactly those from Wang et al.; they handle vertices with multiple degree‑1 neighbours, degree‑2 vertices that belong to a triangle, and degree‑2 vertices that are not part of a triangle. The authors add Rules 4‑7, which target specific configurations involving degree‑3 vertices, degree‑4 vertices with a pendant, and small “diamond‑like” structures. Each rule is proved to be safe: applying it preserves planarity, the existence of a connected vertex cover of size k, and updates the parameter appropriately (often decreasing k by one). After exhausting these rules, the resulting graph G₁ contains no degree‑2 vertices and each vertex has at most one degree‑1 neighbour.

Phase 2 – Face‑based merging of degree‑1 neighbours.
The second phase introduces a novel rule (Rule 8) that merges two degree‑1 neighbours when their incident vertices lie on the same face of a fixed planar embedding. To maximise the number of applications of this rule, the algorithm builds an auxiliary graph G_M whose vertices correspond to the vertices of G₁ that have a degree‑1 neighbour. Two vertices are adjacent in G_M if they share a face in G₁. A maximum matching M₀ in G_M is computed (using the Micali‑Vazirani algorithm). The matching is then “re‑ordered” per face to obtain another matching M of the same size but with the property that the matched edges can be applied simultaneously as Rule 8 operations. Each matched pair reduces the total vertex count by one while preserving planarity and the existence of a CVC of the appropriate size. The time spent in this phase is dominated by the matching computation, which runs in O(n²·⁵) time.

Phase 3 – Bounding the kernel size.
Let (G₂, k₂) be the instance after Phase 2. The authors prove that if G₂ contains a connected vertex cover of size k₂, then |V(G₂)| ≤ (11/3)·k₂. The proof proceeds by considering a minimum connected vertex cover S in G₁ (which is also a minimum cover in G₂ because S contains no degree‑1 vertices). The vertices outside S form an independent set I, which is partitioned into I₁ (degree‑1 vertices), I₃ (degree‑3 vertices), and I_{≥4} (degree at least 4). Standard planar bipartite edge‑count arguments give the crude bound |I| < 3|S|, but this is not tight when the additional reductions are applied. The authors identify three “good events” that shrink I further: a large matching M* (many merges performed in Phase 2), a large set S_{≥3} (many vertices of S have no degree‑1 neighbours), and a large set I_{≥4} (many high‑degree vertices outside S). Lemma 2 shows that at least one of these events must contribute a total of at least |S|/3, i.e., |S_{≥3}| + |I_{≥4}| + |M*| ≥ |S|/3. Lemma 3, a matching‑size bound for planar graphs of minimum degree 3, underpins this claim. Combining these inequalities yields the final bound |V(G₂)| ≤ (11/3)·|S| ≤ (11/3)·k₂.

If after Phase 3 the graph still has more than (11/3)·k₂ vertices, the algorithm safely answers “NO” for the original instance. Thus the algorithm produces a kernel of size at most (11/3)·k, and runs in polynomial time.

Tightness.
The paper concludes with a construction showing that the analysis of Lemma 2 is tight: there exist planar graphs for which the three quantities |S_{≥3}|, |I_{≥4}|, and |M*| each attain roughly |S|/9, making the bound (11/3)·k unavoidable without introducing new reduction rules. This demonstrates that any further improvement must rely on additional structural reductions beyond those presented.

Significance.
Reducing the linear factor from 4 to 11/3 is a substantial improvement in the realm of linear kernels for planar problems, where the constant directly influences the exponent of exponential‑time algorithms that may be applied after kernelization. Moreover, the face‑based matching technique introduced in Phase 2 is a novel tool that could be adapted to other planar parameterized problems. The work therefore advances both the specific state of the art for Connected Vertex Cover and the general methodology for designing tight linear kernels on planar graphs.