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Updated: Jan 17, 2026

Generating Strictly Controlled Stimuli for Figure Recognition Experiments

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On the Parameterized Complexity of Eulerian Strong Component Arc Deletion.

Václav Blažej¹, Satyabrata Jana¹, M S Ramanujan¹

¹University of Warwick, Coventry, UK.

Algorithmica

|September 23, 2025

Summary

This study investigates the Eulerian Strong Component Arc Deletion problem, proving it

Area of Science:

Graph Theory
Computational Complexity

Background:

The Eulerian Strong Component Arc Deletion problem aims to minimize arc deletions in a directed multigraph so that each resulting strongly connected component is Eulerian.
This problem extends the Directed Feedback Arc Set problem and has applications in housing market analysis.
The parameterized complexity of this problem, particularly concerning solution size, was a previously unresolved question.

Purpose of the Study:

To resolve the fixed-parameter tractability of the Eulerian Strong Component Arc Deletion problem parameterized by solution size.
To conduct a comprehensive complexity analysis for various parameterizations, including treewidth and maximum degree.

Main Methods:

Utilized parameterized complexity theory to establish lower bounds on computational complexity.

Keywords:

Eulerian graphs Parameterized complexity Treewidth

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Developed algorithms for specific parameterizations, analyzing their efficiency.

Employed techniques from graph theory and complexity theory to prove hardness and tractability results.

Main Results:

Ruled out a fixed-parameter tractable (FPT) algorithm for the problem when parameterized by solution size, assuming standard complexity conjectures.
Demonstrated W[1]-hardness or para-NP-hardness when parameterized solely by treewidth or maximum degree.
Established XP-completeness for treewidth parameterization and FPT results for combined parameters (treewidth and maximum degree, or treewidth and solution size).

Conclusions:

The Eulerian Strong Component Arc Deletion problem is computationally hard for several natural parameterizations.
Efficient algorithms exist when parameters are combined, offering practical solutions for certain graph structures.
The study provides a complete picture of the problem's parameterized complexity, with implications for related graph problems.