On the Parameterized Complexity of Contraction to Generalization of Trees

Akanksha Agarwal, Saket Saurabh, Prafullkumar Tale

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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 ⊆ 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 ℓ edges. Thus, the problem we study is T -Contraction. We design an FPT algorithm for T -Contraction running in time O((2ℓ+2)O(k+ℓ)⋅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+2ℓ)](⌈αα−1⌉+1)).

Original languageEnglish
Pages (from-to)587-614
Number of pages28
JournalTheory of Computing Systems
Volume63
Issue number3
DOIs
StatePublished - 15 Apr 2019
Externally publishedYes

Keywords

  • Fixed parameter tractability
  • Generalization of trees
  • Graph algorithms
  • Graph contraction

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics

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