Polynomial kernels for Proper Interval Completion and related problems

Given a graph G=(V,E)G=(V,E) and a positive integer k, the Proper Interval Completion problem asks whether there exists a set F of at most k   pairs of (V×V)∖E(V×V)∖E such that the graph H=(V,E∪F)H=(V,E∪F) is a proper interval graph. The Proper Interval Completion problem finds applications in molecular biology and genomic research. This problem is known to be FPT (Kaplan, Tarjan and Shamir, FOCSʼ94), but no polynomial kernel was known to exist. We settle this question by proving that Proper Interval Completion admits a kernel with O(k3)O(k3) vertices. Moreover, we prove that a related problem, the so-called Bipartite Chain Deletion problem, admits a kernel with O(k2)O(k2) vertices, completing a previous result of Guo (ISAACʼ07).