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` be the family of graphs such that each graph in T` can be made into a tree by deleting at most ` edges. Thus, the problem we study is T`-Contraction. We design an FPT algorithm for T`-Contraction running in time O((2 √ ` + 2)O(k+`) · nO(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`-Contraction of size O([k(k + 2`)](d α α−1 e+1)). 1998 ACM Subject Classification G.2.2 Graph Algorithms, I.1.2 Analysis of Algorithms