Paths to trees and cacti

Akanksha Agrawal, Lawqueen Kanesh, Saket Saurabh, Prafullkumar Tale

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

7 Scopus citations

Abstract

For a family of graphs (formula presented), the (formula presented) -Contraction problem takes as an input a graph G and an integer k, and the goal is to decide whether there exists (formula presented) of size at most k such that G/F belongs to (formula presented). When (formula presented) is the family of paths, trees or cacti, then the correspond-ing problems are Path Contraction, Tree Contraction and Cac-tus Contraction, respectively. It is known that Tree Contraction and Cactus Contraction do not admit a polynomial kernel unless (formula presented), while Path Contraction admits a kernel with O(k) vertices. The starting point of this article are the following natural ques-tions: What is the structure of the family of paths that allows Path Con-traction to admit a polynomial kernel? Apart from the size of the solu-tion, what other additional parameters should we consider so that we can design polynomial kernels for these basic contraction problems? With the goal of designing polynomial kernels, we consider the family of trees with bounded number of leaves (note that the family of paths are trees with at most two leaves). In particular, we study Bounded Tree Contraction (Bounded TC). Here, an input is a graph G, integers k and, (formula presented) and the goal is to decide whether or not, there exists a subset (formula presented) of size at most k such that G/F is a tree with at most leaves. We design a kernel for Bounded TC with O(k) vertices and (formula presented) edges. Finally, we study Bounded Cactus Contraction (Bounded CC) which takes as input a graph G and integers k and. The goal is to decide whether there exists a subset (formula presented) of size at most k such that G/F is a cactus graph with at most (formula presented) leaf blocks in the corresponding block decomposi-tion. For Bounded CC we design a kernel with (formula presented) vertices and (formula presented) edges. We complement our results by giving kernelization lower bounds for Bounded TC, Bounded OTC and Bounded CC by showing that unless (formula presented) the size of the kernel we obtain is optimal.

Original languageEnglish
Title of host publicationAlgorithms and Complexity - 10th International Conference, CIAC 2017, Proceedings
EditorsDimitris Fotakis, Aris Pagourtzis, Vangelis Th. Paschos
PublisherSpringer Verlag
Pages31-42
Number of pages12
ISBN (Print)9783319575858
DOIs
StatePublished - 1 Jan 2017
Externally publishedYes
Event10th International Conference on Algorithms and Complexity, CIAC 2017 - Athens, Greece
Duration: 24 May 201726 May 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10236 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference10th International Conference on Algorithms and Complexity, CIAC 2017
Country/TerritoryGreece
CityAthens
Period24/05/1726/05/17

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)

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