Feedback vertex sets in (directed) graphs of bounded degeneracy or treewidth

Kolja Knauer (1, 2) , Xuan Hoang La (3) , Petru Valicov (3)

1 LIS - Laboratoire d'Informatique et des Systèmes (LIS) (Marseille, Toulon)
2 Departament de Matematiques i Informatica, Universitat de Barcelona
3 LIRMM | ALGCO - Algorithmes, Graphes et Combinatoire

We study the minimum size $f$ of a feedback vertex set in directed and undirected $n$-vertex graphs of given degeneracy or treewidth. In the undirected setting the bound $\frac{k-1}{k+1}n$ is known to be tight for graphs with bounded treewidth $k$ or bounded odd degeneracy $k$. We show that neither of the easy upper and lower bounds $\frac{k-1}{k+1}n$ and $\frac{k}{k+2}n$ can be exact for the case of even degeneracy. More precisely, for even degeneracy $k$ we prove that $\frac{3k-2}{3k+4}n\leq f < \frac{k}{k+2}n$. For directed graphs of bounded degeneracy $k$, we prove that $f\leq\frac{k-1}{k+1}n$ and that this inequality is strict when $k$ is odd. For directed graphs of bounded treewidth $k\geq 2$, we show that $\frac{k-2\lfloor\log_2(k)\rfloor}{k+1}n\leq f \leq \frac{k}{k+3}n$. Further, we provide several constructions of low degeneracy or treewidth and large $f$.

Domaines: Informatique [cs]

Kolja Knauer, Xuan Hoang La, Petru Valicov. Feedback vertex sets in (directed) graphs of bounded degeneracy or treewidth. The Electronic Journal of Combinatorics, 2022, 29 (4), ⟨10.37236/10914⟩. ⟨hal-03583220⟩