### Abstract

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.

Original language | English |
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Publisher | ArXiv |

Publication status | Published - 17 Jan 2017 |

### Keywords

- cs.DM
- cs.SI
- math.CO

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## Cite this

Smith, J. P., & Zahedi, E. (2017).

*On Distance Preserving and Sequentially Distance Preserving Graphs*. ArXiv. https://arxiv.org/abs/1701.06404