Distance preserving graphs and graph products

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.

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