TY - GEN

T1 - Algorithms and complexity for geodetic sets on planar and chordal graphs

AU - Chakraborty, Dibyayan

AU - Das, Sandip

AU - Foucaud, Florent

AU - Gahlawat, Harmender

AU - Lajou, Dimitri

AU - Roy, Bodhayan

N1 - Funding Information:
Funding This research was financed by the IFCAM project “Applications of graph homomorphisms” (MA/IFCAM/18/39) and the ANR project HOSIGRA (ANR-17-CE40-0022). Bodhayan Roy: B. Roy is supported by an ISIRD Grant from Sponsored Research and Industrial Consultancy, IIT Kharagpur.
Publisher Copyright:
© Dibyayan Chakraborty, Sandip Das, Florent Foucaud, Harmender Gahlawat, Dimitri Lajou, and Bodhayan Roy.

PY - 2020/12/1

Y1 - 2020/12/1

N2 - 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.

AB - 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.

KW - Chordal graph

KW - FPT algorithm

KW - Geodetic set

KW - Interval graph

KW - Planar graph

UR - http://www.scopus.com/inward/record.url?scp=85100947975&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ISAAC.2020.7

DO - 10.4230/LIPIcs.ISAAC.2020.7

M3 - Conference contribution

AN - SCOPUS:85100947975

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 71

EP - 715

BT - 31st International Symposium on Algorithms and Computation, ISAAC 2020

A2 - Cao, Yixin

A2 - Cheng, Siu-Wing

A2 - Li, Minming

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 31st International Symposium on Algorithms and Computation, ISAAC 2020

Y2 - 14 December 2020 through 18 December 2020

ER -