Tour Theory
Introduction
In graph theory, a walk (in English, walk, and sometimes also translated as tour)[1] is a succession of vertices "Vertex (graph theory)") and edges "Edge (graph theory)") within a graph, which begins and ends at vertices, such that each vertex is incident with the edges that follow and precede it in the sequence.[2] Two vertices are connected or accessible if there is a path that forms a trajectory to get from one to the other; Otherwise, the vertices are disconnected or inaccessible.[1].
Two vertices can be connected by multiple paths. The length of a path is its number of edges. Thus, in an undirected graph, adjacent vertices are connected by a path of length 1, the second neighbors "Neighborhood (graph theory)") by a path of length 2, and so on. An undirected graph is connected if all its vertices are connected through a path.[2] A connected graph whose vertices and edges allow a path to be defined is a path graph.
formal definition
Given a graph, a path is a sequence of vertices and edges such that (in case the graph is undirected), or (in case it is directed), for all . The length of the path is .[2][1].
Types of related trajectories
Contenido
Existen varios conceptos derivados del de camino:[2].
Trajectories in directed graphs
The above definitions of paths also apply to directed graphs, as long as the paths respect the direction of the edges between each vertex and the next. However, if in a directed graph you want to ignore the direction of the edges and consider their trajectories as if it were an undirected graph, then the paths are known as semipaths, the paths as semipaths, the cycles as semicycles, etc.[1].