Parameterized Complexity of Geodetic Set

TL;DR

This paper investigates the parameterized complexity of the Geodetic Set problem, establishing its hardness for some parameters and providing fixed-parameter algorithms for others, advancing understanding of its computational boundaries.

Contribution

It proves W[1]-hardness for certain parameters and develops fixed-parameter algorithms for others, enriching the complexity landscape of Geodetic Set.

Findings

W[1]-hard when parameterized by feedback vertex number, path-width, and combined solution size.

Fixed-parameter algorithms for feedback edge number, tree-depth, and modular-width.

Enhanced understanding of the problem's complexity across different graph parameters.

Abstract

A vertex set $S$ of a graph $G$ is geodetic if every vertex of $G$ lies on a shortest path between two vertices in $S$. Given a graph $G$ and $k \in \mathbb N$, the NP-hard Geodetic Set problem asks whether there is a geodetic set of size at most $k$. Complementing various works on Geodetic Set restricted to special graph classes, we initiate a parameterized complexity study of Geodetic Set and show, on the negative side, that Geodetic Set is W[1]-hard when parameterized by feedback vertex number, path-width, and solution size, combined. On the positive side, we develop fixed-parameter algorithms with respect to the feedback edge number, the tree-depth, and the modular-width of the input graph.