  The isometric path cover (partition) problem of a graph is to find a minimum
set of isometric paths which cover (partition) the vertex set of the graph. The
isometric path cover (partition) number of a graph is the cardinality a minimum
isometric path cover (partition). We prove that the isometric path partition
problem and the isometric $k$-path partition problem for $k\geq 3$ are
NP-complete on general graphs. Fisher and Fitzpatrick \cite{FiFi01} have shown
that the isometric path cover number of $(r\times r)$-dimensional grid is
$\lceil 2r/3\rceil$. We show that the isometric path cover (partition) number
of $(r\times s)$-dimensional grid is $s$ when $r \geq s(s-1)$. We establish
that the isometric path cover (partition) number of $(r\times r)$-dimensional
torus is $r$ when $r$ is even and is either $r$ or $r+1$ when $r$ is odd. Then,
we demonstrate that the isometric path cover (partition) number of an
$r$-dimensional Benes network is $2^r$. In addition, we provide partial
solutions for the isometric path cover (partition) problems for cylinder and
multi-dimensional grids.
