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 - 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 -