We all have played this game in our childhood, remember? But you must be thinking what this game has to do with graph connectivity. Let me answer this question. Connectivity is a concept related to linking. Just as in this game, we link one dot to another to complete a drawing in the same way in a graph, we link one vertex to another so that the graph could be traversed. Therefore what is a connected graph? Connected graph definition can be explained as a fundamental concept in the connectivity graph theory. The graph connectivity determines whether the graph could be traversed or not.
Connectivity Graph Theory
A graph can be a connected graph or a disconnected graph depending upon the topological space. If there is a path between the two vertices of a graph then we call it a connected graph. When a graph has multiple disconnected vertices then we call it a disconnected graph. The sub topics of graph connectivity are edge connectivity and vertex connectivity. Edge connected graph is a graph where the edges are removed to make it disconnected while a vertex conned graph is a graph where the vertices are removed to make it disconnected.
Edge Connectivity In Graph Theory
When the minimum number of edges gets removed to disconnect a graph. It is known as edge connectivity in graph theory. It can be represented as λ(G). Therefore, if λ(G) ≥ k, then the graph G will be called k-edge-connected.
Here is an example edge connectivity in graph theory:
Here, a graph can become disconnected by removing just two minimum edges. Therefore, this will be called a 2-edge-connected graph. There are four ways we can disconnect the graph by removing two edges:
Vertex Connectivity
If we remove the minimum number of vertices from a graph to make it a disconnected graph then it will be called a vertex connectivity in connectivity graph theory. It is represented as K(G). therefore, if K(G) ≥ k, then it will be called vertex connectivity in connectivity graph theory. But in order to disconnect a graph by removing a vertex, we also have to remove the edges incident to it. Here is an example of vertex graph connectivity.
If we remove the vertex c or d, the graph will become disconnected. Since we only need to remove one vertex to disconnect the graph, we call it 1-edge-connected graph.
Cut Vertex
When only one vertex is enough to disconnect a graph it is called a cut-vertex.
If G is a connected graph and a vertex v is removed from G and the graph disconnects then it will be called a cut vertex of G. As a result of removing a vertex from a graph the graph will break into two or more graphs.
In the first diagram, by removing just the vertex c, the graph becomes a disconnected graph.
Cut Edge (Bridge)
A cut Edge is also called bridge and it disconnects the graph by just removing a single edge.
If G is a connected graph and an edge e of the graph G is removed from it to make the graph a disconnected one then we can call it a cut edge graph connectivity. As a result of removing an edge from a graph the graph will break into two or more graphs. This removal of the edge is called a cut edge or bridge.
In the diagram above, by removing edge c and e, the graph becomes a disconnected graph according to the graph connectivity.
Cut Set
A cut set is a set of edges that disconnects a graph only when all the edges are removed. If only a few edges are removed then it does not disconnect the graph. Here is an example:
In this diagram, we disconnect the graph only partially by removing just three edges i.e. bd, be and ce. Therefore, {bd, be, ce} is a cut set.
After removing these edges, the graph connectivity will look like this:
Solved Examples
Question1: Consider a complete graph that has a total of 20 vertices, therefore, find the number of edges it may consist of.
Solution 1: The formula for the total number of edges in a k15 graph can be represented as:
Number of edges = n/2(n-1)
= 20/2(20-1)
=10(19)
=190
Therefore, it consists of 190 edges.
Question 2: If a graph consists of 40 edges, then how many vertices does will it have?
Solution 2: We know that
Number of edges = n/2 (n-1)
40 = n/2 (n-1)
n(n-1) = 80
n2 – n – 80 = 0
If we solve the above quadratic equation, we will get;
n ≈ 9.45, -8.45
Since, the number of vertices cannot be negative.
Therefore, n ≈ 9