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Vertex Deletion into Bipartite Permutation Graphs

Publication at Faculty of Mathematics and Physics |
2020

Abstract

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines 𝓁1 and 𝓁1, 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 [John M.

Lewis and Mihalis Yannakakis, 1980]. 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 f(k)n^O(1), and also give a polynomial-time 9-approximation algorithm.