Edge Exploration of Temporal Graphs

We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast with the static case, it is NP $$\textsc {NP}$$ -hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each...

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Bibliographic Details
Published inCombinatorial Algorithms Vol. 12757; pp. 107 - 121
Main Authors Bumpus, Benjamin Merlin, Meeks, Kitty
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 2021
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Fixed parameter tractability
Temporal Eulerian circuit
Temporal exploration
Temporal graphs
Online AccessGet full text
ISBN9783030799861
3030799867
ISSN0302-9743
1611-3349
1611-3349
DOI10.1007/978-3-030-79987-8_8

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Summary:We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast with the static case, it is NP $$\textsc {NP}$$ -hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each edge is active at only three times. However, we do obtain an FPT $$\textsc {FPT}$$ -algorithm with respect to a new parameter called interval-membership-width which restricts the times assigned to different edges; we believe that this parameter will be of independent interest for other temporal graph problems. Our techniques also allow us to resolve two open questions of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related problem of exploring temporal stars.
Bibliography:Original Abstract: We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast with the static case, it is NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsc {NP}$$\end{document}-hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each edge is active at only three times. However, we do obtain an FPT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsc {FPT}$$\end{document}-algorithm with respect to a new parameter called interval-membership-width which restricts the times assigned to different edges; we believe that this parameter will be of independent interest for other temporal graph problems. Our techniques also allow us to resolve two open questions of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related problem of exploring temporal stars.
B. M. Bumpus—Supported by an EPSRC doctoral training account.K. Meeks—Supported by a Royal Society of Edinburgh Personal Research Fellowship, funded by the Scottish Government, and EPSRC grant EP/T004878/1.
ISBN:9783030799861
3030799867
ISSN:0302-9743
1611-3349
1611-3349
DOI:10.1007/978-3-030-79987-8_8