A Polynomial Kernel for Bipartite Permutation Vertex Deletion

Abstract

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 $S\subseteq V(G)$ such that $|S| \le k$ 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 ${\mathcal {O}}(9^k |V(G)|^{9})$. 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 $k'\le k$, and $|V(G')|+|E(G')|\le k^{\mathcal {O}(1)}$.