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Vertex Deletion into Bipartite Permutation Graphs.

Łukasz Bożyk1, Jan Derbisz2, Tomasz Krawczyk2

  • 1Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland.

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Summary
This summary is machine-generated.

This study investigates the bipartite permutation vertex deletion problem, aiming to find algorithms for efficiently identifying graphs that can become bipartite permutation graphs by removing few vertices. Researchers developed a fixed-parameter tractable algorithm and a 9-approximation algorithm.

Keywords:
Comparability graphsGraph modification problemsPartially ordered setPermutation graphs

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Area of Science:

  • Graph Theory
  • Computational Complexity
  • Discrete Mathematics

Background:

  • Permutation graphs are intersection graphs of line segments.
  • Bipartite permutation graphs are a subclass of permutation graphs that are also bipartite.
  • The bipartite permutation vertex deletion problem is NP-complete.

Purpose of the Study:

  • To analyze the parameterized complexity of the bipartite permutation vertex deletion problem.
  • To develop efficient algorithms for this problem.

Main Methods:

  • Analyzing the structure of almost bipartite permutation graphs.
  • Exploiting the properties of the shortest hole in such graphs.
  • Developing a fixed-parameter tractable algorithm and a polynomial-time approximation algorithm.

Main Results:

  • An algorithm with running time O(2^k * n^c) for the bipartite permutation vertex deletion problem.
  • A 9-approximation algorithm for the same problem.

Conclusions:

  • The structural properties of holes in almost bipartite permutation graphs are key to solving the problem.
  • Efficient algorithms, both exact (parameterized) and approximate, can be developed for this computationally hard problem.