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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 )).
A graph is called k-vertex-connected or k-connected if its vertex connectivity is k or greater. More precisely, any graph G (complete or not) is said to be k -vertex-connected if it contains at least k + 1 vertices, but does not contain a set of k − 1 vertices whose removal disconnects the graph; and κ ( G ) is defined as the largest k such ...
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 ...
A strongly connected graph is a directed graph in which every ordered pair of vertices in the graph is strongly connected. Otherwise, it is called a weakly connected graph if every ordered pair of vertices in the graph is weakly connected. Otherwise it is called a disconnected graph. A k-vertex-connected graph or k-edge-connected graph is a ...
The algorithm for weak components generates the strongly connected components in this order, and maintains a partition of the components that have been generated so far into the weak components of their induced subgraph. After all components are generated, this partition will describe the weak components of the whole graph. [2] [3]
In graph theory, Robbins' theorem, named after Herbert Robbins (), states that the graphs that have strong orientations are exactly the 2-edge-connected graphs.That is, it is possible to choose a direction for each edge of an undirected graph G, turning it into a directed graph that has a path from every vertex to every other vertex, if and only if G is connected and has no bridge.
weakly connected A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. weight A numerical value, assigned as a label to a vertex or edge of a graph. The weight of a subgraph is the sum of the weights of the vertices or edges within that subgraph ...
A graph with three components. In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting ...