Further Kernelization of Proper Interval Vertex Deletion: New Observations and Refined Analysis
Wenjun Li, Yongjie Yang, Jianer Chen, Jianxin Wang

TL;DR
This paper presents a significant reduction in the kernel size for the Proper Interval Vertex Deletion problem, decreasing it from a large polynomial to a more manageable size through new insights and refined analysis.
Contribution
The authors develop an $O(k^7)$-vertex kernel for PIVD, improving previous bounds with new observations and a refined analytical approach.
Findings
Achieved an $O(k^7)$-vertex kernel for PIVD.
Introduced new observations to improve kernelization.
Provided a refined analysis leading to a smaller kernel.
Abstract
In the Proper Interval Vertex Deletion problem (PIVD for short), we are given a graph and an integer parameter , and the question is whether there are at most vertices in whose removal results in a proper interval graph. It is known that the PIVD problem is fixed-parameter tractable and admits a polynomial but "unreasonably" large kernel of vertices. A natural question is whether the problem admits a polynomial kernel of "reasonable" size. In this paper, we answer this question by deriving an -vertex kernel for the PIVD problem. Our kernelization is based on several new observations and a refined analysis of the kernelization.
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Optimization and Search Problems
