Generalized Pseudoforest Deletion: Algorithms and Uniform Kernel

Abstract

Feedback Vertex Set (FVS) is one of the most well studied problems in the realm of parameterized complexity. In this problem we are given a graph G and a positive integer k and the objective is to test whether there exists $S\subseteq V(G)$ of size at most k such that $G-S$ is a forest. Thus, FVS is about deleting as few vertices as possible to get a forest. The main goal of this paper is to study the following interesting problem: How can we generalize the family of forests such that the nice structural properties of forests and the interesting algorithmic properties of FVS can be extended to problems on this class? Towards this we define a graph class, $\mathcal{F}_l$, that contains all graphs where each connected component can transformed into forest by deleting at most l edges. The class $\mathcal{F}_1$ is known as pseudoforest in the literature and we call $\mathcal{F}_l$ as l-pseudoforest. We study the problem of deleting k-vertices to get into $\mathcal{F}_l$, $l$-pseudoforest Deletion, in the realm of parameterized complexity. We show that $l$-pseudoforest Deletion admits an algorithm with running time $c_l^k n^{{\mathcal {O}}(1)}$ and admits a kernel of size $f(l)k^2$. Thus, for every fixed l we have a kernel of size ${\mathcal {O}}(k^2)$. That is, we get a uniform polynomial kernel for $l$-pseudoforest Deletion. For the special case of $l=1$, we design an algorithm with running time $7.5618^k n^{{\mathcal {O}}(1)}$. Our algorithms and uniform kernels combine iterative compression, expansion lemma and protrusion machinery.