Lecture
A connected graph is a graph containing exactly one connected component. This means that there is at least one path between any pair of vertices of this graph.
A direct application of graph theory is the theory of networks — and its application — the theory of electronic networks. For example, all computers connected to the Internet form a connected graph, and although a single pair of computers may not be connected directly (in the wording for graphs, not be connected by an edge), you can transfer information to any other computer from any computer vertices of the graph in any other).
In oriented graphs, several concepts of connectedness are distinguished.
An oriented graph is said to be strongly connected if there exists a (oriented) path from any vertex to any other, or, equivalently, the graph contains exactly one strongly connected component.
A directed graph is said to be weakly connected if the unoriented graph obtained from it is replaced by a replacement of oriented edges with an undirected one.
Here are some criterion (equivalent) definitions of a connected graph:
A graph is said to be simply connected (connected) if:
A directed graph is called strongly connected or strong if for any two different vertices Xi and Xj there is at least one path connecting Xi with XjThis definition also means that any two vertices of such a graph are mutually reachable.
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Discrete Math. Set theory. Graph theory. Combinatorics.
Terms: Discrete Math. Set theory. Graph theory. Combinatorics.