Modular Decomposition of Graphs and the Distance Preserving Property

Given a graph G, a subgraph H is isometric if d_H (u,v) = d_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.