Distance Preserving Graphs

Given a graph $G$ then a subgraph $H$ is $isometric$ if, for every pair of vertices $u,v$ of $H$, we have $d_H(u,v) = d_G(u,v)$. 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 consider how to add a vertex to a dp graph so that the result is a dp graph. This condition implies that chordal graphs are dp. We also find a condition on the girth of $G$ which implies that it is not dp. In closing, we discuss other work and open problems concerning dp graphs.