Connectivity
ExampleIn the above example, it is possible to travel from one vertex to another vertex. Here, we can traverse from vertex B to H using the path B > A > D > F > E > H. Hence it is a connected graph. ExampleIn the above example, it is not possible to traverse from vertex B to H because there is no path between them directly or indirectly. Hence, it is a disconnected graph. Let's see some basic concepts of Connectivity. 1. Cut VertexA single vertex whose removal disconnects a graph is called a cutvertex. Let G be a connected graph. A vertex v of G is called a cut vertex of G, if Gv (Remove v from G) results a disconnected graph. When we remove a vertex from a graph then graph will break into two or more graphs. This vertex is called a cut vertex. Note: Let G be a graph with n vertices:
Example 1Example 2In the above graph, vertex 'e' is a cutvertex. After removing vertex 'e' from the above graph the graph will become a disconnected graph. 2. Cut Edge (Bridge)A cut Edge or bridge is a single edge whose removal disconnects a graph. Let G be a connected graph. An edge e of G is called a cut edge of G, if Ge (Remove e from G) results a disconnected graph. When we remove an edge from a graph then graph will break into two or more graphs. This removal edge is called a cut edge or bridge. Note: Let G be a graph with n vertices:
Example 1In the above graph, edge (c, e) is a cutedge. After removing this edge from the above graph the graph will become a disconnected graph. Example 2In the above graph, edge (c, e) is a cutedge. After removing this edge from the above graph the graph will become a disconnected graph. 3. Cut SetIn a connected graph G, a cut set is a set S of edges with the following properties:
Example 1To disconnect the above graph G, we have to remove the three edges. i.e. bd, be and ce. We cannot disconnect it by removing just two of three edges. Hence, {bd, be, ce} is a cut set. After removing the cut set from the above graph, it would look like as follows: 4. Edge ConnectivityThe edge connectivity of a connected graph G is the minimum number of edges whose removal makes G disconnected. It is denoted by λ(G). When λ(G) ≥ k, then graph G is said to be kedgeconnected. ExampleLet's see an example, From the above graph, by removing two minimum edges, the connected graph becomes disconnected graph. Hence, its edge connectivity is 2. Therefore the above graph is a 2edgeconnected graph. Here are the following four ways to disconnect the graph by removing two edges: 5. Vertex ConnectivityThe connectivity (or vertex connectivity) of a connected graph G is the minimum number of vertices whose removal makes G disconnects or reduces to a trivial graph. It is denoted by K(G). The graph is said to be k connected or kvertex connected when K(G) ≥ k. To remove a vertex we must also remove the edges incident to it. ExampleLet's see an example: The above graph G can be disconnected by removal of the single vertex either 'c' or 'd'. Hence, its vertex connectivity is 1. Therefore, it is a 1connected graph.
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