Graph Theory - Connectivity

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Connectivity of a Graph

The connectivity of a graph refers to the extent to which the graph remains connected when vertices or edges are removed. A graph is said to be connected if there is a path between any two vertices in the graph.

Types of Connectivity

There are two main types of connectivity in a graph −

• Vertex Connectivity: The minimum number of vertices that need to be removed to disconnect the graph or make it disconnected.
• Edge Connectivity: The minimum number of edges that need to be removed to disconnect the graph or make it disconnected.

Calculating Vertex Connectivity

To calculate the vertex connectivity, we follow these steps −

• Find all the subsets of vertices whose removal disconnects the graph.
• Identify the smallest subset of vertices that disconnects the graph.
• The size of this smallest subset is the vertex connectivity.

Example: Vertex Connectivity

Consider the following graph −

If we remove vertex A, the graph becomes disconnected. Therefore, the vertex connectivity is 1, as removing a single vertex disconnects the graph.

Calculating Edge Connectivity

To calculate the edge connectivity, we follow these steps −

• Identify all subsets of edges whose removal disconnects the graph.
• Find the smallest subset of edges that disconnects the graph.
• The size of this smallest subset is the edge connectivity.

Example: Edge Connectivity

Consider the following graph −

If we remove the edge (A, B), the graph becomes disconnected. Therefore, the edge connectivity is 1, as removing a single edge disconnects the graph.

Types of Connected Graphs

Graphs can be classified into different types based on their connectivity, such as −

• Connected Graph: A graph is said to be connected if there is a path between every pair of vertices in the graph.
• Disconnected Graph: A graph is disconnected if it is not connected, meaning there is at least one pair of vertices that are not connected by any path.

Connected Graph

A graph is connected if there exists a path between every pair of vertices. In such graphs, it is possible to reach any vertex from any other vertex.

In this graph, there is a path between every pair of vertices, so it is a connected graph.

Disconnected Graph

A disconnected graph consists of two or more components that are not connected by any path. These components are isolated from each other.

In this graph, there are two disconnected components, so the graph is considered disconnected.

Connectivity of Tree

A tree is a special type of graph that is connected and acyclic. It has exactly one path between any two vertices. Removing any vertex or edge from a tree will disconnect the graph, making its connectivity properties quite strong.

In this tree, removing any edge disconnects the graph, and it remains connected without any cycles.

Connectivity of Complete Graph

A complete graph is a graph in which there is an edge between every pair of vertices. It is maximally connected, and its vertex and edge connectivity are both equal to the number of vertices minus one.

In this graph, there is an edge between every pair of vertices, making it a complete graph with high connectivity.

Connectivity of Bipartite Graph

A bipartite graph is a graph whose vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent. Bipartite graphs are often used in modeling relationships between two distinct sets of objects.

This graph is bipartite because its vertices can be divided into two sets, with edges only connecting vertices between sets.

Connectivity in Various Graphs

Here are examples of the connectivity of different types of graphs −

Graph Type	Vertex Connectivity	Edge Connectivity
Path Graph	1 Removing any vertex disconnects the graph	1 Removing any edge disconnects the graph
Star Graph	1 Removing the center vertex disconnects the graph	1 Removing any edge disconnects the graph
Wheel Graph	2 Removing two non-adjacent vertices disconnects the graph	2 Removing two non-adjacent edges disconnects the graph
Complete Graph	n1 Removing n1 vertices disconnects the graph	n1 Removing n1 edges disconnects the graph

Importance of Connectivity

Connectivity plays an important role in various applications of graph theory, such as −

• Network Design: Ensuring that a network is connected is crucial for communication between different parts of the system.
• Reliability Analysis: Graph connectivity helps to determine how strong a system is to failures, such as the removal of nodes or edges.
• Routing Algorithms: Connectivity is important for finding the shortest path or optimal routing in transportation or communication networks.