Path (graph theory)
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See  e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs. Definitions. Walk, trail, and path. If is a finite walk with vertex sequence then "w" is said to be a "walk from" "v" "to" "v". Similarly for a trail or a path. If there is a finite walk between two "distinct" vertices then there is also a finite trail and a finite path between them. Some authors do not require that all vertices of a path be distinct and instead use the term simple path to refer  to such a path where all vertices are distinct. A weighted graph associates a value ("weight") with every edge in the graph. The "weight of a walk" (or trail or path) in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words "cost" or "length" are used instead of weight. Directed walk, directed trail, and directed path. If is a finite directed walk with vertex sequence then "w" is said to be a "walk from" "v" "to" "v". Similarly for a directed trail or a path. If there is a finite directed walk between  two "distinct" vertices then there is also a finite directed trail and a finite directed path between them. Some authors do not require that all vertices of a directed path be distinct and instead use the term simple directed path to refer to such a directed path. A weighted directed graph associates a value ("weight") with every edge in the directed graph. The "weight of a directed walk" (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words "cost" or "length" are used instead of weight. Finding paths . Several algorithms exist to find shortest and longest paths in graphs, with the important distinction that the former problem is computationally much easier than the latter. Dijkstra's algorithm produces a list of shortest paths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (or no edge weights), whilst the Bellman–Ford algorithm can be applied to directed graphs with negative edge weights. The Floyd–Warshall algorithm can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.