# Modular Decomposition of Graphs and the Distance Preserving Property

Research output: Working paper

### Abstract

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.
Original language English ArXiv Published - 24 May 2018

### Fingerprint

Modular Decomposition
Graph in graph theory
Isometric
Subgraph
Lexicographic Product
Cartesian product
Distance Function
Divides
Disjoint

• cs.DM
• math.CO

### Cite this

ArXiv, 2018.

Research output: Working paper

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