On the computational complexity of the strong geodetic recognition problem

Abstract

A vertex set $S$ of a graph $G$ is a strong geodetic set if it is possible to choose exactly one shortest path for each pair of vertices in $S$, and cover all the remaining vertices of the graph with these paths. In the \textsc{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-complete for general graphs. In this work we introduce the \textsc{Strong Geodetic Recognition} problem (SGR), which consists in determining whether 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.