Modular Decomposition of Graphs and the Distance Preserving Property

TL;DR

This paper introduces a new graph decomposition method based on modular decomposition and lexicographic product generalization, providing a characterization of distance preserving graphs and their properties under graph products.

Contribution

It presents a novel generalization of the lexicographic product, linking modular decomposition to distance preserving properties in graphs.

Findings

Characterization of distance preserving graphs using modular decomposition

Introduction of a generalized lexicographic product for graph analysis

Proof that the Cartesian product of a dp and an sdp graph is dp

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.