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The {claw, diamond}-free edge deletion problem admits an \(O(k^3)\) -vertex kernel [Theoretical Computer Science 2022]. In this paper, we improve it to \(O(k^2)\) vertices. We use the modulator technique and then bound the number of vertices outside the modulator. Let M be an arbitrary modulator. We find that, the propagation between the maximal cliques in \(G-M\) is always due to the claw graph. Based on the observation, the number of vertices in the maximal cliques untouched to M can be bounded by \(O(k^2)\) . Since \(|M|=O(k)\) and the maximal cliques in \(G-M\) touched to M is bounded by some normal kernel frameworks, we thus have the claimed kernelization size.

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A Quadratic Kernel for {Claw, Diamond}-free Deletion

  • 会议论文

  • 发表日期：

    2026年

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  • Hanchun Yuan

摘要

The {claw, diamond}-free edge deletion problem admits an \(O(k^3)\) -vertex kernel [Theoretical Computer Science 2022]. In this paper, we improve it to \(O(k^2)\) vertices. We use the modulator technique and then bound the number of vertices outside the modulator. Let M be an arbitrary modulator. We find that, the propagation between the maximal cliques in \(G-M\) is always due to the claw graph. Based on the observation, the number of vertices in the maximal cliques untouched to M can be bounded by \(O(k^2)\) . Since \(|M|=O(k)\) and the maximal cliques in \(G-M\) touched to M is bounded by some normal kernel frameworks, we thus have the claimed kernelization size.