On the Parameterized Complexity of Contraction to Generalization of Trees

Abstract

For a family of graphs $\mathcal {F}$ , 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 ⊆ E(G) of size at most k such that G/S belongs to $\mathcal {F}$ . 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 $\mathcal {F}$ -Contraction when $\mathcal {F}$ is a simple family of graphs such as trees and paths. In this paper, we study the $\mathcal {F}$ -Contraction problem, where $\mathcal {F}$ 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 l edges. Thus, the problem we study is $\mathbb {T}_{\ell }$ -Contraction. We design an FPT algorithm for $\mathbb {T}_{\ell }$ -Contraction running in time $\mathcal {O}((2\sqrt {\ell }+ 2)^{\mathcal {O}(k + \ell )} \cdot n^{\mathcal {O}(1)})$ . 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)}})$ .