  If $G$ is a graph then a subgraph $H$ is $isometric$ if, for every pair of
vertices $u,v$ of $H$, we have $d_H(u,v) = d_G(u,v)$ where $d$ is the distance
function. We say a graph $G$ is $distance\ preserving\ (dp)$ if it has an
isometric subgraph of every possible order up to the order of $G$. We give a
necessary and sufficient condition for the lexicographic product of two graphs
to be a dp graph. A graph $G$ is $sequentially\ distance\ preserving\ (sdp)$ if
the vertex set of $G$ can be ordered so that, for all $i\ge1$, deleting the
first $i$ vertices in the sequence results in an isometric graph. We show that
the Cartesian product of two graphs is sdp if and only if each of them is sdp.
In closing, we state a conjecture concerning the Cartesian products of dp
graphs.
