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Check if a graph is strongly connected | Set 1 (Kosaraju using DFS)

A graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected. In an undirected graph G , two vertices u and v are called connected if G contains a path from u to v.

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Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length 1 , i.

A graph is said to be connected if every pair of vertices in the graph is connected. A connected component is a maximal connected subgraph of G. Each vertex belongs to exactly one connected component, as does each edge. A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected undirected graph. It is unilaterally connected or unilateral if it contains a directed path from u to v or a directed path from v to u for every pair of vertices u, v.

Check if a graph is strongly connected | Set 1 (Kosaraju using DFS) - GeeksforGeeks

The strong components are the maximal strongly connected subgraphs. A cut , vertex cut , or separating set of a connected graph G is a set of vertices whose removal renders G disconnected.

Graph Theory - An Introduction!

A graph is called k -connected or k -vertex-connected if its vertex connectivity is k or greater. A vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v. A graph G which is connected but not 2 -connected is sometimes called separable. Analogous concepts can be defined for edges.

From Wikipedia, the free encyclopedia

In the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a bridge. More generally, an edge cut of G is a set of edges whose removal renders the graph disconnected. Again, local edge-connectivity is symmetric. A graph is called k -edge-connected if its edge connectivity is k or greater.

Distance connectivity in graphs and digraphs

A graph is said to be maximally connected if its connectivity equals its minimum degree. A graph is said to be maximally edge-connected if its edge-connectivity equals its minimum degree.

Distance connectivity in graphs and digraphs

One of the most important facts about connectivity in graphs is Menger's theorem , which characterizes the connectivity and edge-connectivity of a graph in terms of the number of independent paths between vertices. If u and v are vertices of a graph G , then a collection of paths between u and v is called independent if no two of them share a vertex other than u and v themselves.

Similarly, the collection is edge-independent if no two paths in it share an edge. The problem of determining whether two vertices in a graph are connected can be solved efficiently using a search algorithm , such as breadth-first search.

More generally, it is easy to determine computationally whether a graph is connected for example, by using a disjoint-set data structure , or to count the number of connected components. A simple algorithm might be written in pseudo-code as follows:. In computational complexity theory, SL is the class of problems log-space reducible to the problem of determining whether two vertices in a graph are connected, which was proved to be equal to L by Omer Reingold in The problem of computing the probability that a Bernoulli random graph is connected is called network reliability and the problem of computing whether two given vertices are connected the ST-reliability problem.

Both of these are P -hard. The first few non-trivial terms are. From Wikipedia, the free encyclopedia. This article needs additional citations for verification. Now it is possible to determine the biggest connected group and the size of it. Also you can see with who a certain individual is connected. Learn About Live Editor. Choose a web site to get translated content where available and see local events and offers.

Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance. Other MathWorks country sites are not optimized for visits from your location. Toggle Main Navigation. Search MathWorks. File Exchange. Open Mobile Search. Trial software. You are now following this Submission You will see updates in your activity feed You may receive emails, depending on your notification preferences. Connectivity check for undirected graphs. For an adjacency matrix it checks if the vertices are connected with each other.

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