Parameterized Eulerian Strong Component Arc Deletion Problem on Tournaments

TL;DR

This paper investigates the parameterized complexity of the { extsc Min-DESC} problem on tournaments, proving it is fixed-parameter tractable when parameterized by the number of arcs removed.

Contribution

The paper establishes that the { extsc Min-DESC} problem on tournaments is fixed-parameter tractable, answering an open question about its complexity.

Findings

{ extsc Min-DESC} is NP-hard in general.

The problem is fixed-parameter tractable on tournaments.

The result advances understanding of arc deletion problems in directed graphs.

Abstract

In the problem {\sc Min-DESC}, we are given a digraph $D$ and an integer $k$, and asked if there exists a set $A'$ of at most $k$ arcs in $D$, such that if we remove the arcs of $A'$, in the resulting digraph every strong component is Eulerian. {\sc Min-DESC} is NP-hard; Cechl\'{a}rov\'{a} and Schlotter (IPEC 2010) asked if the problem is fixed-parameter tractable when parameterized by $k$. We consider the subproblem of{\sc Min-DESC} when $D$ is a tournament. We show that this problem is fixed-parameter tractable with respect to $k$.