Graph Hamiltonicity Parameterized by Proper Interval Deletion Set

Petr A. Golovach, R. Krithika, Abhishek Sahu, Saket Saurabh, Meirav Zehavi

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

Abstract

The Path Cover and Cycle Cover problems are well-known generalizations of the classical Hamiltonian Path and Hamiltonian Cycle problems. Here, we are given an undirected graph on n vertices and a positive integer r and the task is to check if there are r vertex-disjoint paths (cycles) that together visit all the vertices of the graph exactly once. Path Cover and Cycle Cover remain NP-hard even when restricted to chordal graphs (Information Processing Letters 1986) but are polynomial-time solvable on proper interval graphs (Discrete Mathematics 1993 and Proceedings of WADS 2019). In this paper, we study the complexity of Path Cover and Cycle Cover with respect to a structural parameter, namely, distance to proper interval graphs. In particular, we show that Path Cover and Cycle Cover are fixed-parameter tractable (FPT) when parameterized by k, the size of a proper interval deletion set (a set of vertices whose deletion results in a proper interval graph). For this purpose, we design an algorithm with O(2O ( k log k )nO ( 1 )) running time for each of these problems. Our algorithms use several interesting properties of proper interval graphs and a dynamic programming procedure over clique partitions to solve these problems in the mentioned time. As a consequence, we get the same fixed-parameter tractability results for Hamiltonian Cycle and Hamiltonian Path problems with the same parameterization. Recently, Chaplick et al. (Proceedings of WADS 2019) obtained polynomial kernels and compression algorithms for Path Cover and Cycle Cover parameterized by a different measure of similarity with proper interval graphs. Our FPT algorithms also adds to this study of structural parameterizations for these classical problems.

Original languageEnglish
Title of host publicationLATIN 2020
Subtitle of host publicationTheoretical Informatics - 14th Latin American Symposium 2021, Proceedings
EditorsYoshiharu Kohayakawa, Flávio Keidi Miyazawa
PublisherSpringer Science and Business Media Deutschland GmbH
Pages104-115
Number of pages12
ISBN (Print)9783030617912
DOIs
StatePublished - 1 Jan 2020
Event14th Latin American Symposium on Theoretical Informatics, LATIN 2020 - Sao Paulo, Brazil
Duration: 5 Jan 20218 Jan 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12118 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference14th Latin American Symposium on Theoretical Informatics, LATIN 2020
Country/TerritoryBrazil
CitySao Paulo
Period5/01/218/01/21

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