Further Kernelization of Proper Interval Vertex Deletion: New Observations and Refined Analysis

In the Proper Interval Vertex Deletion problem (PIVD), we are given a graph G and an integer parameter k > 0, and the question is whether there are at most k vertices in G whose removal results in a proper interval graph. It is known that the PIVD problem is fixedparameter tractable and admits a polynomial but “unreasonably” large kernel of O(k53) 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 O(k7)-vertex kernel for the PIVD problem. Our kernelization is based on several new observations and a refined analysis of the kernelization. 1998 ACM Subject Classification F.2.2 Nonnumerical algorithms and problems