On Distance Preserving and Sequentially Distance Preserving Graphs

A graph H is an isometric subgraph of G if dH(u, v) = dG(u, v), for every pair u, v ∈ V (H). A graph is distance preserving if it has an isometric subgraph of every possible order. A graph is sequentially distance preserving if its vertices can be ordered such that deleting the first i vertices results in an isometric subgraph, for all i ≥ 1. 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.