Vertex deletion into bipartite permutation graphs

TL;DR

This paper investigates the problem of deleting vertices to transform a graph into a bipartite permutation graph, providing an FPT algorithm and a polynomial approximation, advancing understanding of the structural complexity of these graphs.

Contribution

It introduces a fixed-parameter tractable algorithm and a polynomial approximation for the bipartite permutation vertex deletion problem, based on structural analysis of almost bipartite permutation graphs.

Findings

Developed an $O(9^k imes n^9)$ algorithm for the problem.

Proposed a polynomial-time 9-approximation algorithm.

Analyzed the structure of holes in almost bipartite permutation graphs.

Abstract

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $l_1$ and $l_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, 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 by the classical result of Lewis and Yannakakis. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time $O(9^k\cdot n^9)$, and also give a polynomial-time 9-approximation algorithm.