### Abstract

Original language | English |
---|---|

Publisher | ArXiv |

Publication status | Published - 24 May 2018 |

### Fingerprint

### Keywords

- cs.DM
- math.CO

### Cite this

*Modular Decomposition of Graphs and the Distance Preserving Property*. ArXiv.

**Modular Decomposition of Graphs and the Distance Preserving Property.** / Zahedi, Emad; Smith, Jason P.

Research output: Working paper

}

TY - UNPB

T1 - Modular Decomposition of Graphs and the Distance Preserving Property

AU - Zahedi, Emad

AU - Smith, Jason P.

N1 - 11 pages

PY - 2018/5/24

Y1 - 2018/5/24

N2 - Given a graph $G$, a subgraph $H$ is isometric if $d_H(u,v) = d_G(u,v)$ for every pair $u,v\in V(H)$, 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 $i\ge1$. 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.

AB - Given a graph $G$, a subgraph $H$ is isometric if $d_H(u,v) = d_G(u,v)$ for every pair $u,v\in V(H)$, 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 $i\ge1$. 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.

KW - cs.DM

KW - math.CO

M3 - Working paper

BT - Modular Decomposition of Graphs and the Distance Preserving Property

PB - ArXiv

ER -