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Domination and Cut Problems on Chordal Graphs with Bounded Leafage

Authors Esther Galby, Dániel Marx , Philipp Schepper , Roohani Sharma , Prafullkumar Tale

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Author Details

Esther Galby
  • TU Hamburg, Germany
Dániel Marx
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Philipp Schepper
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Roohani Sharma
  • Max Planck Institute for Informatics, SIC, Saarbrücken, Germany
Prafullkumar Tale
  • Indian Institute of Science Education and Research, Pune, India

Funding

Research supported by the European Research Council (ERC) consolidator grant No. 725978 SYSTEMATICGRAPH. Esther Galby and Prafullkumar Tale: Part of the work was carried out when the authors were Post-Doctoral Researchers at CISPA Helmholtz Center for Information Security, Germany.
  • Schepper, Philipp: Part of Saarbrücken Graduate School of Computer Science, Germany.

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Esther Galby, Dániel Marx, Philipp Schepper, Roohani Sharma, and Prafullkumar Tale. Domination and Cut Problems on Chordal Graphs with Bounded Leafage. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 14:1-14:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.IPEC.2022.14

Abstract

The leafage of a chordal graph G is the minimum integer 𝓁 such that G can be realized as an intersection graph of subtrees of a tree with 𝓁 leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time 2^𝒪(𝓁²) ⋅ n^𝒪(1). We present a conceptually much simpler algorithm that runs in time 2^𝒪(𝓁) ⋅ n^𝒪(1). We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple n^𝒪(𝓁)-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in n^O(1)-time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Chordal Graphs
  • Leafage
  • FPT Algorithms
  • Dominating Set
  • MultiCut with Undeletable Terminals
  • Multiway Cut with Undeletable Terminals

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