Feedback Vertex Sets in Tournaments

TL;DR

This paper investigates the number and enumeration of minimal feedback vertex sets in tournament graphs, providing new bounds and an efficient enumeration algorithm, advancing both combinatorial understanding and algorithmic techniques.

Contribution

It establishes new upper and lower bounds on the number of minimal feedback vertex sets and introduces the first polynomial space, polynomial delay enumeration algorithm.

Findings

Maximum of 1.6740^n minimal feedback vertex sets in any n-vertex tournament

Existence of tournaments with at least 1.5448^n minimal feedback vertex sets

Fastest known algorithm for minimum feedback vertex set in tournaments

Abstract

We study combinatorial and algorithmic questions around minimal feedback vertex sets in tournament graphs. On the combinatorial side, we derive strong upper and lower bounds on the maximum number of minimal feedback vertex sets in an n-vertex tournament. We prove that every tournament on n vertices has at most 1.6740^n minimal feedback vertex sets, and that there is an infinite family of tournaments, all having at least 1.5448^n minimal feedback vertex sets. This improves and extends the bounds of Moon (1971). On the algorithmic side, we design the first polynomial space algorithm that enumerates the minimal feedback vertex sets of a tournament with polynomial delay. The combination of our results yields the fastest known algorithm for finding a minimum size feedback vertex set in a tournament.