On the Minimum Cycle Cover problem on graphs with bounded co-degeneracy

TL;DR

This paper introduces a fixed-parameter tractable algorithm for the Minimum Cycle Cover problem on graphs with bounded co-degeneracy, extending previous results related to Hamiltonian cycles and graph parameters.

Contribution

It determines the stability of the cycle cover property and develops a $2^{O(k)} ext{poly}(n)$-time algorithm for graphs with co-degeneracy at most $k$, generalizing prior work.

Findings

Proves the stability of the bounded cycle cover property.

Provides a fixed-parameter algorithm for graphs with bounded co-degeneracy.

Extends previous algorithms for Hamiltonian Cycle to cycle cover problem.

Abstract

In 2021, Duarte, Oliveira, and Souza [MFCS 2021] showed some problems that are FPT when parameterized by the treewidth of the complement graph (called co-treewidth). Since the degeneracy of a graph is at most its treewidth, they also introduced the study of co-degeneracy (the degeneracy of the complement graph) as a parameter. In 1976, Bondy and Chv\'{a}tal [DM 1976] introduced the notion of closure of a graph: let $\ell$ be an integer; the $(n+\ell)$-closure, $\operatorname{cl}_{n+\ell}(G)$, of a graph $G$ with $n$ vertices is obtained from $G$ by recursively adding an edge between pairs of nonadjacent vertices whose degree sum is at least $n+\ell$ until no such pair remains. A graph property $\Upsilon$ defined on all graphs of order $n$ is said to be $(n+\ell)$-stable if for any graph $G$ of order $n$ that does not satisfy $\Upsilon$, the fact that $uv$ is not an edge of $G$ and that $G+uv$ satisfies $\Upsilon$ implies $d(u)+d(v)< n+\ell$. Duarte et al. [MFCS 2021] developed an algorithmic framework for co-degeneracy parameterization based on the notion of closures for solving problems that are $(n+\ell)$-stable for some $\ell$ bounded by a function of the co-degeneracy. In this paper, we first determine the stability of the property of having a bounded cycle cover. After that, combining the framework of Duarte et al. [MFCS 2021] with some results of Jansen, Kozma, and Nederlof [WG 2019], we obtain a $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$-time algorithm for Minimum Cycle Cover on graphs with co-degeneracy at most $k$, which generalizes Duarte et al. [MFCS 2021] and Jansen et al. [WG 2019] results concerning the Hamiltonian Cycle problem.