 License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2017.1
URN: urn:nbn:de:0030-drops-85446
URL: https://drops.dagstuhl.de/opus/volltexte/2018/8544/
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### On the Parameterized Complexity of Contraction to Generalization of Trees

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### Abstract

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 \subseteq 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 \cal 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_\ell be the family of graphs such that each graph in T_\ell can be made into a tree by deleting at most \ell edges. Thus, the problem we study is T_\ell-Contraction. We design an FPT algorithm for T_\ell-Contraction running in time O((\ncol)^{O(k + \ell)} * 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_\ell-Contraction of size O([k(k + 2\ell)] ^{(\lceil {\frac{\alpha}{\alpha-1}\rceil + 1)}}).

### BibTeX - Entry

@InProceedings{agrawal_et_al:LIPIcs:2018:8544,
author =	{Akanksha Agrawal and Saket Saurabh and Prafullkumar Tale},
title =	{{On the Parameterized Complexity of Contraction to Generalization of Trees}},
booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
pages =	{1:1--1:12},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-051-4},
ISSN =	{1868-8969},
year =	{2018},
volume =	{89},
editor =	{Daniel Lokshtanov and Naomi Nishimura},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},

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