On the Minimum Cycle Cover Problem on Graphs with Bounded Co-degeneracy

Abstract

AbstractIn 2017, Knop, Koutecký, Masařík, and Toufar [WG 2017] asked about the complexity of deciding graph problems \(\varPi \) on the complement of G considering a parameter p of G, especially for sparse graph parameters such as treewidth. 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 Chvátal [DM 1976] introduced the notion of closure of a graph: let \(\ell \) be an integer; the \((n+\ell )\)-closure, \({\text {cl}}_{n+\ell }(G)\), 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 \(\varUpsilon \) 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 \(\varUpsilon \), the fact that uv is not an edge of G and that \(G+uv\) satisfies \(\varUpsilon \) 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 2019, Jansen, Kozma, and Nederlof [WG 2019] relax the conditions of Dirac’s theorem and consider input graphs G in which at least \(n-k\) vertices have degree at least \(\frac{n}{2}\), and present an FPT algorithm concerning to k, to decide whether such graphs G are Hamiltonian. 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 et al. [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.KeywordsDegeneracyComplement graphCycle coverClosureFPTKernel