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An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or ray) has a first vertex but no last vertex. 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]
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.
An extension of Robbins' theorem to mixed graphs by Boesch & Tindell (1980) shows that, if G is a graph in which some edges may be directed and others undirected, and G contains a path respecting the edge orientations from every vertex to every other vertex, then any undirected edge of G that is not a bridge may be made directed without changing the connectivity of G.
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).
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).
For each , the directed nature of provides for a natural indexing of its vertices from the start to the end of the path. For each vertex v {\displaystyle v} in G i {\displaystyle G_{i}} , we locate the first vertex in Q {\displaystyle Q} reachable by v {\displaystyle v} , and the last vertex in Q {\displaystyle Q} that reaches to v ...
a is inserted between v 2 and v 3.. Any tournament on a finite number of vertices contains a Hamiltonian path, i.e., directed path on all vertices (Rédei 1934).. This is easily shown by induction on : suppose that the statement holds for , and consider any tournament on + vertices.
Clipping path. By convention, the inside of the path is defined by its direction. Reversing the direction of a path reverses what is considered inside or outside. An inclusive path is one where what is visually "inside" the path corresponds to what will be preserved; an exclusive path, of opposite direction, contains what is visually "outside ...