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If G is connected then its line graph L(G) is also connected. A graph G is 2-edge-connected if and only if it has an orientation that is strongly connected. Balinski's theorem states that the polytopal graph (1-skeleton) of a k-dimensional convex polytope is a k-vertex-connected graph. [13]
The strongly connected components of a directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ( V + E )).
Tarjan's strongly connected components algorithm is an algorithm in graph theory for finding the strongly connected components (SCCs) of a directed graph. It runs in linear time , matching the time bound for alternative methods including Kosaraju's algorithm and the path-based strong component algorithm .
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). The ...
The edge relation [note 1] of a tournament graph is always a connected relation on the set of ' s vertices. If a strongly connected relation is symmetric, it is the universal relation. A relation is strongly connected if, and only if, it is connected and reflexive. [proof 1]
A totally cyclic orientation of a graph G is an orientation in which each edge belongs to a directed cycle. For connected graphs, this is the same thing as a strong orientation, but totally cyclic orientations may also be defined for disconnected graphs, and are the orientations in which each connected component of G becomes strongly connected ...
It is the conceptually simplest efficient algorithm, but is not as efficient in practice as Tarjan's strongly connected components algorithm and the path-based strong component algorithm, which perform only one traversal of the graph. If the graph is represented as an adjacency matrix, the algorithm requires Ο(V 2) time.
A directed graph is strongly connected if it contains a directed path from every vertex to every other vertex. Then we have the following theorem (Bang-Jensen & Gutin 2007, Theorem 7.2.2): A directed graph is strongly connected if and only if it has an ear decomposition.