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On Graphs Coverable by k Shortest Paths

Authors Maël Dumas, Florent Foucaud , Anthony Perez, Ioan Todinca

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Author Details

Maël Dumas
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France
Florent Foucaud
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
Anthony Perez
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France
Ioan Todinca
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France

Funding

  • Foucaud, Florent: This author was financed by the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39), the ANR project GRALMECO (ANR-21-CE48-0004-01) and the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25).

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Maël Dumas, Florent Foucaud, Anthony Perez, and Ioan Todinca. On Graphs Coverable by k Shortest Paths. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ISAAC.2022.40

Abstract

We show that if the edges or vertices of an undirected graph G can be covered by k shortest paths, then the pathwidth of G is upper-bounded by a function of k. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph G and a set of k pairs of vertices called terminals, asks whether G can be covered by k shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph G and a set of k terminals, asks whether there exist binom(k,2) shortest paths, each joining a distinct pair of terminals such that these paths cover G). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter k.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Shortest paths
  • covering problems
  • parameterized complexity

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