Vertex Deletion into Bipartite Permutation Graphs

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $$\ell _1$$ℓ 1 and $$\ell _2$$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 $$\mathsf {NP}$$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 $${\mathcal {O}}(9^k \cdot n^9)$$O ( 9 k · n 9 ) , and also give a polynomial-time 9-approximation algorithm.