A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $\ell_1$ and $\ell_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In the the bipartite permutation vertex deletion problem we ask for a given $n$-vertex graph, whether we can remove at most $k$ vertices to obtain a bipartite permutation graph. This problem is NP-complete but it does admit an FPT algorithm parameterized by $k$. In this paper we study the kernelization of this problem and show that it admits a polynomial kernel with $O(k^{62})$ vertices.

翻译：排列图可定义为端点在两条平行直线$\ell_1$和$\ell_2$上的线段（每条直线上各有一个端点）的交图。二分排列图是同时为二分图的排列图。在二分排列图顶点删除问题中，给定一个$n$顶点图，我们询问是否可以通过删除至多$k$个顶点得到一个二分排列图。该问题是NP完全的，但存在参数化为$k$的FPT算法。本文研究该问题的核化性质，并证明其具有包含$O(k^{62})$个顶点的多项式核。