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Covering and Partitioning of Split, Chain and Cographs with Isometric Paths

Authors Dibyayan Chakraborty , Haiko Müller , Sebastian Ordyniak , Fahad Panolan , Mateusz Rychlicki

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Author Details

Dibyayan Chakraborty
  • School of Computing, University of Leeds, UK
Haiko Müller
  • School of Computing, University of Leeds, UK
Sebastian Ordyniak
  • School of Computing, University of Leeds, UK
Fahad Panolan
  • School of Computing, University of Leeds, UK
Mateusz Rychlicki
  • School of Computing, University of Leeds, UK

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Dibyayan Chakraborty, Haiko Müller, Sebastian Ordyniak, Fahad Panolan, and Mateusz Rychlicki. Covering and Partitioning of Split, Chain and Cographs with Isometric Paths. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.39

Abstract

Given a graph G, an isometric path cover of a graph is a set of isometric paths that collectively contain all vertices of G. An isometric path cover 𝒞 of a graph G is also an isometric path partition if no vertex lies in two paths in 𝒞. Given a graph G, and an integer k, the objective of Isometric Path Cover (resp. Isometric Path Partition) is to decide whether G has an isometric path cover (resp. partition) of cardinality k. 
In this paper, we show that Isometric Path Partition is NP-complete even on split graphs, i.e. graphs whose vertex set can be partitioned into a clique and an independent set. In contrast, we show that both Isometric Path Cover and Isometric Path Partition admit polynomial time algorithms on cographs (graphs with no induced P₄) and chain graphs (bipartite graphs with no induced 2K₂).

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Isometric path partition (cover)
  • chordal graphs
  • chain graphs
  • split graphs

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References

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