Edge, vertex expansion, contraction from "summary" of Introduction to Graph Theory by Douglas Brent West
Graphs are an important tool for understanding the relationships between different elements, and edge, vertex expansion, and contraction are techniques used to modify the structure of a graph. These techniques add or remove edges to increase or reduce the number of vertices in the graph, allowing us to explore the structure of the graph in more detail.- Edge expansion is the process of adding an edge between two vertices, connecting them. This increases the involved number ofadjacencies to two or more and hence gives rise to greater connectivity within the graph.
- Forced by the fact that any two vertices of a graph can be connected through some path, vertex contraction deletes vertices from the graph. Thus, its effect on the graph depends upon the paths that lead to those vertices. When contracting certain pairs of vertices, interesting configurations may be developed within such graphs.
- Degree distribution is critical in assessing the effectiveness of vertex expansion or contraction since it measures how many connections exist. Editing individual vertices can affect whole degrees of neighboring vertices eventually impacting subsequent station characteristics like propagation, throughput etc.
- While extremely useful, experimentation with edge addition or deletion should not be overdone as it can have unintended results such as shortening the reachability radius or the average edge length of the graph.
- Ultimately, operations such as edge expansion, contraction and vertex interact heavily in transforming regular lattice configuration into small world topology characterized by high degree of locality embedded inside larger networks of turbulences.
