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Parameterized Complexity of Eulerian Deletion Problems.

Marek Cygan1, Dániel Marx2, Marcin Pilipczuk1

  • 1Institute of Informatics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland.

Algorithmica
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Summary
This summary is machine-generated.

This study classifies the complexity of making graphs Eulerian by vertex or edge deletions. Edge-deletion problems are efficiently solvable, unlike node-deletion problems which are computationally hard.

Keywords:
Deletion distanceEulerian graphFixed-parameter tractabilityKernelization

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Area of Science:

  • Graph Theory
  • Computational Complexity Theory
  • Algorithms

Background:

  • Making a graph Eulerian (connected with all even degrees) is a fundamental problem.
  • Understanding the computational cost of graph modification problems is crucial for algorithm design.

Purpose of the Study:

  • To classify the parameterized complexity of making graphs Eulerian through minimum deletions.
  • To differentiate the complexity between vertex and edge deletion variants.

Main Methods:

  • Parameterized complexity analysis.
  • W-hardness proofs for intractable cases.
  • Fixed-parameter tractable (FPT) algorithm design using color coding for edge deletions.

Main Results:

  • Node-deletion problems for making graphs Eulerian are W[1]-hard across all studied variants.
  • Edge-deletion problems are either polynomial-time solvable or fixed-parameter tractable.
  • A novel randomized FPT algorithm is presented for undirected graphs using edge deletions.

Conclusions:

  • The complexity landscape for making graphs Eulerian differs significantly between vertex and edge deletion.
  • Edge-deletion problems offer efficient algorithmic solutions, while node-deletion problems remain computationally challenging.
  • Polynomial kernelization is not possible for the NP-complete but FPT variants.