On the Computational Complexity of the Strong Geodetic Recognition Problem

TL;DR

This paper studies the computational complexity of recognizing strong geodetic sets in graphs, proving NP-completeness for the recognition problem and providing complexity results for specific graph classes.

Contribution

It introduces the Strong Geodetic Recognition problem, proves its NP-completeness, and analyzes its complexity across various graph classes, including chordal graphs.

Findings

SGR is NP-complete.

Polynomial algorithms for certain graph classes.

Resolved open question for chordal graphs.

Abstract

A strong geodetic set of a graph~$G=(V,E)$ is a vertex set~$S \subseteq V(G)$ in which it is possible to cover all the remaining vertices of~$V(G) \setminus S$ by assigning a unique shortest path between each vertex pair of~$S$. In the Strong Geodetic problem (SG) a graph~$G$ and a positive integer~$k$ are given as input and one has to decide whether~$G$ has a strong geodetic set of cardinality at most~$k$. This problem is known to be NP-hard for general graphs. In this work we introduce the Strong Geodetic Recognition problem (SGR), which consists in determining whether even a given vertex set~$S \subseteq V(G)$ is strong geodetic. We demonstrate that this version is NP-complete. We investigate and compare the computational complexity of both decision problems restricted to some graph classes, deriving polynomial-time algorithms, NP-completeness proofs, and initial parameterized complexity results, including an answer to an open question in the literature for the complexity of SG for chordal graphs.