On the Parameterized Complexity of Contraction to generalization of trees

For a family of graphs F, the F-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists S ⊆ E(G) of size at most k such that G/S belongs to 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 F-Contraction when F is a simple family of graphs such as trees and paths. In this paper, we study the F-Contraction problem, where F generalizes the family of trees. In particular, we define this generalization in a "parameterized way". Let T_ℓ be the family of graphs such that each graph in T_ℓ can be made into a tree by deleting at most ℓ edges. Thus, the problem we study is T_ℓ-Contraction. We design an FPT algorithm for T^ℓ-Contraction running in time O((2√ℓ + 2)^O(k+ℓ) · n^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 T_ℓ-Contraction of size O([k(k + 2ℓ)]^{([α/α-1]+1)}).