Algorithms and complexity for geodetic sets on interval and chordal graphs

2026-06-29

AI Summary

This study investigates the computational complexity of the minimum geodetic set problem on chordal and interval graphs, which seeks the smallest vertex subset such that every vertex in the graph lies on a shortest path between two vertices in the subset. The authors resolve an open question posed by Ekim et al. by proving, for the first time, that the problem is NP-hard on interval graphs. Furthermore, they develop a polynomial-time algorithm for chordal graphs when the treewidth is fixed, leveraging parameterized complexity analysis and treewidth theory. By combining a refined reduction exploiting the linear structure of interval graphs with insights from treewidth-based dynamic programming, the work precisely delineates the complexity boundary of the problem on these two important graph classes.

Abstract

We study the computational complexity of finding the geodetic number of a graph on chordal graphs and interval graphs. A set $S$ of vertices of a graph $G$ is a \textit{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. We show that \textsc{Minimum Geodetic Set} is fixed parameter tractable for chordal graphs 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{Minimum Geodetic Set} is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012), who showed that \textsc{Minimum Geodetic Set} is polynomial-time solvable on proper interval graphs. 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.

Problem

• geodetic set
• interval graphs
• chordal graphs
• computational complexity
• NP-hard

Innovation

• geodetic set
• chordal graphs
• interval graphs
• fixed-parameter tractability
• NP-hardness