Vertex edge connected from "summary" of Introduction to Graph Theory by Douglas Brent West
A graph is said to be vertex edge connected if for every pair of vertices in the graph, there is a path between them that contains at least one edge. In other words, a graph is vertex edge connected if it is possible to travel from any vertex to any other vertex by following a sequence of adjacent vertices connected by edges. This concept is important in understanding the connectivity of a graph and the relationships between its vertices and edges. Vertex edge connectivity is closely related to the concept of connectivity in graph theory. A graph is said to be connected if there is a path between every pair of vertices in the graph. Vertex edge connectivity takes this idea a step further by requiring that the path between two vertices contains at least one edge. This additional condition provides a more stringent measure of connectivity in a graph. To determine whether a graph is vertex edge connected, one must examine the structure of the graph and the paths that can be formed between its vertices. If it is possible to find a path between every pair of vertices that contains at least one edge, then the graph is vertex edge connected. If there exists a pair of vertices for which no such path can be found, then the graph is not vertex edge connected. Vertex edge connectivity is an important property of graphs in various applications, such as network design, transportation planning, and social network analysis. Understanding the connectivity of a graph can help in analyzing its structure, identifying key vertices and edges, and predicting how information or resources may flow through the graph. By studying the concept of vertex edge connectivity, one can gain valuable insights into the relationships and interactions within a graph.
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