Algorithms and complexity for geodetic sets on planar and chordal graphs - Archive ouverte HAL
Conference Papers Year : 2020

Algorithms and complexity for geodetic sets on planar and chordal graphs

Abstract

We study the complexity of finding the \emph{geodetic number} on subclasses of planar graphs and chordal graphs. A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{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 \textsc{MGS} on restricted classes of planar graphs: we design a linear-time algorithm for \textsc{MGS} on solid grids, improving on a $3$-approximation algorithm by Chakraborty et al. (CALDAM, 2020) and show that it 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 \textsc{MGS} is fixed parameter tractable for inputs of this class when parameterized by its \emph{tree-width} (which equals its clique number). This implies a polynomial-time algorithm for $k$-trees, for fixed $k$. Then, we show that \textsc{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.

Dates and versions

hal-03041361 , version 1 (04-12-2020)

Identifiers

Cite

Dibyayan Chakraborty, Sandip Das, Florent Foucaud, Harmender Gahlawat, Dimitri Lajou, et al.. Algorithms and complexity for geodetic sets on planar and chordal graphs. 31st International Symposium on Algorithms and Computation (ISAAC 2020), Dec 2020, Hong-Kong, Hong Kong SAR China. pp.7:1-15, ⟨10.4230/LIPIcs.ISAAC.2020.7⟩. ⟨hal-03041361⟩
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