Paths sequences connected vertices from "summary" of Introduction to Graph Theory by Douglas Brent West
A path in a graph is a sequence of distinct vertices in which consecutive vertices are adjacent. For example, in the graph G = (V, E), a path from vertex u to vertex v is a finite sequence of distinct vertices v0, v1, v2, ..., vk such that v0 = u, vk = v, and each vi is adjacent to vi+1 for i = 0, 1, 2, ..., k - 1. The length of a path is the number of edges in the path, which is one less than the number of vertices. A sequence of vertices {v0, v1, v2, ..., vk} is a path if and only if {vi, vi+1} is an edge for i = 0, 1, 2, ..., k - 1. The vertices v0 and vk are the endpoints of the path. If the graph is directed, then the edges {vi, vi+1} must be directed edges with vi as the tail and vi+1 as the head. Two vertices u and v are connected if there is a path from u to v. In other words, there is a sequence of vertices v0, v1, v2, ..., vk such that v0 = u, vk = v, and {vi, vi+1} is an edge for i = 0, 1, 2, ..., k - 1. If there is a path from u to v, then v is reachable from u, and u is reachable to v. The connectivity of a graph measures how well connected the vertices are in the graph. A graph is connected if every pair of vertices is connected by a path. A graph is disconnected if there are vertices that are not connected by a path. In a connected graph, there is a path between any two vertices. Paths, sequences, and connected vertices are fundamental concepts in graph theory that help us understand the structure and relationships within a graph. By examining paths and connections between vertices, we can uncover important properties and characteristics of graphs that are essential in various applications and theoretical studies.
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