# On Distance Preserving and Sequentially Distance Preserving Graphs

## TL;DR

This paper investigates the properties of distance preserving and sequentially distance preserving graphs, providing conditions for these properties, especially in graphs without long induced cycles and those with cut vertices.

## Contribution

It offers an equivalent condition for sequentially distance preserving graphs based on simplicial orderings and proves that certain cycle-free graphs are distance preserving.

## Key findings

- Graphs without induced cycles of length ≥5 are sequentially distance preserving.
- Such graphs are also distance preserving.
- A family of non-distance preserving graphs constructed from cycles is defined.

## 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.