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 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 mathsf {NP}-complete by the classical result of Lewis and Yannakakis [20]. 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 {mathcal {O}}(9^k cdot n^9), and also give a polynomial-time 9-approximation algorithm.
Highlights
Many standard computational problems, including maximum clique, maximum independent set, or minimum coloring, which are NP-hard in general, have polynomial-time exact or approximation algorithms in restricted classes of graphs
In this paper we investigate for the first time the modification problems in graph classes related to partial orders
Our main result says that the bipartite permutation vertex deletion problem is fixed parameter tractable
Summary
Many standard computational problems, including maximum clique, maximum independent set, or minimum coloring, which are NP-hard in general, have polynomial-time exact or approximation algorithms in restricted classes of graphs. A linear order (W ,
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