A Polynomial Kernel for Bipartite Permutation Vertex Deletion

In a permutation graph, vertices represent the elements of a permutation, and edges represent pairs of elements that are reversed by the permutation. In the PERMUTATION VERTEX DELETION problem, given an undirected graph G and an integer k, the objective is to test whether there exists a vertex subset S subset of V (G) such that vertical bar S vertical bar <= k and G - S is a permutation graph. The parameterized complexity of PERMUTATION VERTEX DELETION is a well-known open problem. Bozyk et al. [IPEC 2020] initiated a study on this problem by requiring that G - S be a bipartite permutation graph (a permutation graph that is bipartite). They called this the BIPARTITE PERMUTATION VERTEX DELETION (BPVD) problem. They showed that the problem admits a factor 9-approximation algorithm as well as a fixed parameter tractable (FPT) algorithm running in time O (9(k) vertical bar V (G)vertical bar(9)). Moreover, they posed the question whether BPVD admits a polynomial kernel. We resolve this question in the affirmative by designing a polynomial kernel for BPVD. In particular, we obtain the following: Given an instance (G, k) of BPVD, in polynomial time we obtain an equivalent instance (G', k') of BPVD such that k' <= k, and vertical bar V (G')vertical bar + vertical bar E(G')vertical bar <= k(O(1)).