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⊆ V(G) such that |S| ≤ k and G-S is a permutation graph. The parameterized complexity of Permutation Vertex Deletion is a well-known open problem. Bożyk 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 |V(G)|^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 |V(G')|+|E(G')|≤ k^𝒪(1) .