Theory of Computing Systems

, Volume 63, Issue 3, pp 587–614

# On the Parameterized Complexity of Contraction to Generalization of Trees

• Akanksha Agarwal
• Saket Saurabh
• Prafullkumar Tale
Article

## 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 SE(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 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 $$\mathcal {O}([k(k + 2\ell )]^{(\lceil {\frac {\alpha }{\alpha -1}\rceil + 1)}})$$.

## Keywords

Graph contraction Fixed parameter tractability Graph algorithms Generalization of trees

## Notes

### Acknowledgments

A preliminary version of this manuscript has been accepted at the 12th International Symposium on Parameterized and Exact Computation (IPEC 2017).

The research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreements no. 306992 (PARAPPROX).

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