Eulerian Walks in Temporal Graphs

An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal g...

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Vydáno v:Algorithmica Ročník 85; číslo 3; s. 805 - 830
Hlavní autoři: Marino, Andrea, Silva, Ana
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.03.2023
Springer Nature B.V
Témata:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Constraints
Data Structures and Information Theory
Mathematics of Computing
Polynomials
Public transportation
Special Issue on Combinatorial Algorithms (IWOCA 2021)
Theory of Computation
Transportation networks
Temporal graphs
Eulerian trails
Bounded treewidth
Edge cover
Parameterized complexity
Eulerian walks
ISSN:0178-4617, 1432-0541
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Shrnutí:An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal graph ( G , λ ) , with λ : E ( G ) → 2 [ τ ] , an edge e ∈ E ( G ) is available only at the times specified by λ ( e ) ⊆ [ τ ] , in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether ( G , λ ) has a temporal walk or trail is polynomial, while deciding whether it has a local trail is NP -complete even if τ = 2 . In contrast, in the general case, solving any of these problems is NP -complete, even under very strict hypotheses. We finally give XP algorithms parametrized by τ for walks, and by τ + t w ( G ) for trails and local trails, where t w ( G ) refers to the treewidth of G .
Bibliografie:ObjectType-Article-1
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-022-01021-y