### Abstract

Given a graph

*G*, a subgraph*H*is*isometric*if*d*= d_{H}(u,v)_{G}*(u,v)*for every pair*u,v €, V (H)*, where*d*is the distance function. A graph*G*is*distance preserving (dp)*if it has an isometric subgraph of every possible order. A graph is*sequentially distance preserving (sdp)*if its vertices can be ordered such that deleting the first*i*vertices results in an isometric subgraph, for all*i ≥ 1*. We introduce a generalisation of the lexicographic product of graphs, which can be used to non-trivially describe graphs. This generalisation is the inverse of the modular decomposition of graphs, which divides the graph into disjoint clusters called*modules*. Using these operations, we give a necessary and sufficient condition for graphs to be dp. Finally, we show that the Cartesian product of a dp graph and an sdp graph is dp.Original language | English |
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Pages (from-to) | 192-198 |

Number of pages | 7 |

Journal | Discrete Applied Mathematics |

Early online date | 16 May 2019 |

DOIs | |

Publication status | Published - 31 Jul 2019 |

### Keywords

- Distance peserving
- Isometric
- Modular decoposition
- Lexicographic product
- Cartesian product