  A graph $H$ is an \emph{isometric} subgraph of $G$ if $d_H(u,v)= d_G(u,v)$,
for every pair~$u,v\in V(H)$. A graph is \emph{distance preserving} if it has
an isometric subgraph of every possible order. A graph is \emph{sequentially
distance preserving} if its vertices can be ordered such that deleting the
first $i$ vertices results in an isometric subgraph, for all $i\ge1$. We give
an equivalent condition to sequentially distance preserving based upon
simplicial orderings. Using this condition, we prove that if a graph does not
contain any induced cycles of length~$5$ or greater, then it is sequentially
distance preserving and thus distance preserving. Next we consider the distance
preserving property on graphs with a cut vertex. Finally, we define a family of
non-distance preserving graphs constructed from cycles.
