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A directed trail is a directed walk in which all edges are distinct. [2] A directed path is a directed trail in which all vertices are distinct. [2] If w = (e 1, e 2, …, e n − 1) is a finite directed walk with vertex sequence (v 1, v 2, …, v n) then w is said to be a walk from v 1 to v n. Similarly for a directed trail or a path.
A directed graph is weakly connected (or just connected [9]) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph. A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices (x, y).
directed line See arrow. directed path A path in which all the edge s have the same direction. If a directed path leads from vertex x to vertex y, x is a predecessor of y, y is a successor of x, and y is said to be reachable from x. direction 1. The asymmetric relation between two adjacent vertices in a graph, represented as an arrow. 2.
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
A directed acyclic graph is a directed graph that has no cycles. [1] [2] [3] A vertex v of a directed graph is said to be reachable from another vertex u when there exists a path that starts at u and ends at v. As a special case, every vertex is considered to be reachable from itself (by a path with zero edges).
Shortest path (A, C, E, D, F) between vertices A and F in the weighted directed graph. In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding ...
Longest paths of directed acyclic graphs may also be applied in layered graph drawing: assigning each vertex v of a directed acyclic graph G to the layer whose number is the length of the longest path ending at v results in a layer assignment for G with the minimum possible number of layers.