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URN: urn:nbn:de:0030-drops-105874
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### Path Contraction Faster Than 2^n

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### Abstract

A graph G is contractible to a graph H if there is a set X subseteq E(G), such that G/X is isomorphic to H. Here, G/X is the graph obtained from G by contracting all the edges in X. For a family of graphs F, the F-Contraction problem takes as input a graph G on n vertices, and the objective is to output the largest integer t, such that G is contractible to a graph H in F, where |V(H)|=t. When F is the family of paths, then the corresponding F-Contraction problem is called Path Contraction. The problem Path Contraction admits a simple algorithm running in time 2^n * n^{O(1)}. In spite of the deceptive simplicity of the problem, beating the 2^n * n^{O(1)} bound for Path Contraction seems quite challenging. In this paper, we design an exact exponential time algorithm for Path Contraction that runs in time 1.99987^n * n^{O(1)}. We also define a problem called 3-Disjoint Connected Subgraphs, and design an algorithm for it that runs in time 1.88^n * n^{O(1)}. The above algorithm is used as a sub-routine in our algorithm for Path Contraction.

### BibTeX - Entry

```@InProceedings{agrawal_et_al:LIPIcs:2019:10587,
author =	{Akanksha Agrawal and Fedor V. Fomin and Daniel Lokshtanov and Saket Saurabh and Prafullkumar Tale},
title =	{{Path Contraction Faster Than 2^n}},
booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
pages =	{11:1--11:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-109-2},
ISSN =	{1868-8969},
year =	{2019},
volume =	{132},
editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},