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.
Similar Posts
IP addresses identify devices on a network
When devices connect to a network, they need a way to communicate with each other. This is where IP addresses come into play. A...
Growing interconnectedness is inevitable
The idea of growing interconnectedness is not just a trend or a passing phase, but rather an inherent aspect of the evolution o...
IP addresses uniquely identify devices on a network
When devices are connected to a network, they are assigned a unique identifier known as an IP address. This IP address serves a...
Maximum flows model network capacities
The concept of maximum flows model network capacities is a fundamental idea in graph theory. In a network, edges are associated...
Directed graphs model relationships
Directed graphs are a fundamental concept in graph theory that play a crucial role in modeling various relationships. In a dire...
Subgraphs preserve properties graphs
When investigating a graph, one common approach is to examine its subgraphs. Subgraphs are essentially smaller graphs that can ...