On the Parameterized Complexity of Contraction to Generalization of Trees

For a family of graphs $\cal F$, the $\mathcal{F}$-Contraction problem takes as an input a graph $G$ and an integer $k$, and the goal is to decide if there exists $S \subseteq E(G)$ of size at most $k$ such that $G/S$ belongs to $\cal F$. Here, $G/S$ is the graph obtained from $G$ by contracting all the edges in $S$. Heggernes et al.~[Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied $\cal F$-Contraction when $\cal F$ is a simple family of graphs such as trees and paths. In this paper, we study the $\mathcal{F}$-Contraction problem, where $\cal F$ generalizes the family of trees. In particular, we define this generalization in a "parameterized way". Let $\mathbb{T}\ell$ be the family of graphs such that each graph in $\mathbb{T}\ell$ can be made into a tree by deleting at most $\ell$ edges. Thus, the problem we study is $\mathbb{T}\ell$-Contraction. We design an FPT algorithm for $\mathbb{T}\ell$-Contraction running in time $\mathcal{O}((2\sqrt(\ell))^{{\mathcal{O}(k} + \ell)} \cdot n^{{\mathcal{O}(1)})$.} Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by $k$. Inspired by the negative result for the kernelization, we design a lossy kernel for $\mathbb{T}_\ell$-Contraction of size $ \mathcal{O}([k(k + 2\ell)] ^{(\lceil {\frac{\alpha}{\alpha-1}\rceil + 1)}})$.