Algorithms and complexity for geodetic sets on planar and chordal graphs

Dibyayan Chakraborty, Sandip Das, Florent Foucaud, Harmender Gahlawat, Dimitri Lajou, Bodhayan Roy

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations

Abstract

We study the complexity of finding the geodetic number on subclasses of planar graphs and chordal graphs. A set S of vertices of a graph G is a geodetic set if every vertex of G lies in a shortest path between some pair of vertices of S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality of a given graph. The problem is known to remain NP-hard on bipartite graphs, chordal graphs, planar graphs and subcubic graphs. We first study MGS on restricted classes of planar graphs: we design a linear-time algorithm for MGS on solid grids, improving on a 3-approximation algorithm by Chakraborty et al. (CALDAM, 2020) and show that MGS remains NP-hard even for subcubic partial grids of arbitrary girth. This unifies some results in the literature. We then turn our attention to chordal graphs, showing that MGS is fixed parameter tractable for inputs of this class when parameterized by their treewidth (which equals the clique number minus one). This implies a linear-time algorithm for k-trees, for fixed k. Then, we show that MGS is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012). As interval graphs are very constrained, to prove the latter result we design a rather sophisticated reduction technique to work around their inherent linear structure.

Original languageEnglish
Title of host publication31st International Symposium on Algorithms and Computation, ISAAC 2020
EditorsYixin Cao, Siu-Wing Cheng, Minming Li
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages71-715
Number of pages645
ISBN (Electronic)9783959771733
DOIs
StatePublished - 1 Dec 2020
Externally publishedYes
Event31st International Symposium on Algorithms and Computation, ISAAC 2020 - Virtual, Hong Kong, China
Duration: 14 Dec 202018 Dec 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume181
ISSN (Print)1868-8969

Conference

Conference31st International Symposium on Algorithms and Computation, ISAAC 2020
Country/TerritoryChina
CityVirtual, Hong Kong
Period14/12/2018/12/20

Keywords

  • Chordal graph
  • FPT algorithm
  • Geodetic set
  • Interval graph
  • Planar graph

ASJC Scopus subject areas

  • Software

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