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Given a graph G = (V,E) and a positive integer k, the Proper IntervalCompletion problem asks whether there exists a set F of at most k pairs of (V\times V)\E such that the graph H = (V,E \cup F) is a proper interval graph.The Proper Interval Completion problem finds applications in molecular biologyand genomic research. First announced by Kaplan, Tarjan and Shamir in FOCS '94,this problem is known to be FPT, but no polynomial kernel was known to exist.We settle this question by proving that Proper Interval Completion admits akernel with at most O(k^5) vertices. Moreover, we prove that a related problem,the so-called Bipartite Chain Deletion problem, admits a kernel with at mostO(k^2) vertices, completing a previous result of Guo.

Polynomial kernels for Proper Interval Completion and related problems