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In this paper, we study the computational complexity of finding the\emph{geodetic number} of graphs. A set of vertices $S$ of a graph $G$ is a\emph{geodetic set} if any vertex of $G$ lies in some shortest path betweensome pair of vertices from $S$. The \textsc{Minimum Geodetic Set (MGS)} problemis to find a geodetic set with minimum cardinality. In this paper, we provethat solving the \textsc{MGS} problem is NP-hard on planar graphs with amaximum degree six and line graphs. We also show that unless $P=NP$, there isno polynomial time algorithm to solve the \textsc{MGS} problem withsublogarithmic approximation factor (in terms of the number of vertices) evenon graphs with diameter $2$. On the positive side, we give an$O\left(\sqrt[3]{n}\log n\right)$-approximation algorithm for the \textsc{MGS}problem on general graphs of order $n$. We also give a $3$-approximationalgorithm for the \textsc{MGS} problem on the family of solid grid graphs whichis a subclass of planar graphs.

Hardness and approximation for the geodetic set problem in some graph classes