On the Geodomatic Number of a Graph

A set \( S \) of vertices of a graph \( G \) is geodetic if every vertex in \( V(G) \setminus S \) is contained in a shortest path between two vertices of \( S \). The geodetic number \( g(G) \) is the minimum cardinality of a geodetic set of \( G \). The geodomatic number \( d_g(G) \) of a graph \( G \) is the maximum number of elements in a partition of \( V(G) \) into geodetic sets.

In this paper, we determine \( d_g(G) \) for some family of graphs, and we present different bounds on \( d_g(G) \). In particular, we prove the following Nordhaus-Gaddum inequality, where \( \overline{G} \) is the complement of the graph \( G \). If \( G \) is a graph of order \( n \geq 2 \), then
\[
d_g(G) + d_g(\overline{G}) \leq n
\]
with equality if and only if \( n = 2 \).