A polynomial kernel for vertex deletion into bipartite permutation graphs

TL;DR

This paper studies the vertex deletion problem to obtain bipartite permutation graphs, proving it admits a polynomial kernel with size polynomial in the parameter k, despite being NP-complete.

Contribution

It establishes the first polynomial kernel for the bipartite permutation vertex deletion problem, with a kernel size of O(k^{62}).

Findings

The problem is NP-complete but fixed-parameter tractable.

A polynomial kernel with O(k^{62}) vertices is constructed.

This advances kernelization understanding for permutation graph problems.

Abstract

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.