Graph Theory - Edge Connectivity

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Edge Connectivity of a Graph

Edge connectivity of a graph refers to the minimum number of edges that must be removed to disconnect the graph. In other words, it measures how many edges must be removed to make the graph disconnected, or to separate it into two or more disconnected components.

• If a graph is disconnected, its edge connectivity is defined as 0 because no edges need to be removed to disconnect it.
• If a graph is connected, the edge connectivity is at least 1 because removing no edges will leave it connected.
• The higher the edge connectivity, the more strong the graph is to edge removals without losing connectivity.

The edge connectivity of a graph G, denoted by (G), is the smallest number of edges that need to be removed from the graph to either −

• Disconnect the graph (split it into two or more disconnected components).
• Make the graph trivial (a single edge or no edges at all).

Calculating Edge Connectivity

To calculate 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.

Consider the following graph −

In this graph, if we remove the edge AB, the graph becomes disconnected, with vertices C and D separated from the rest. Therefore, the edge connectivity is 1, as removing a single edge disconnects the graph.

Properties of Edge Connectivity

The edge connectivity of a graph has several important properties, such as −

• Non-negative: The edge connectivity of any graph is always non-negative.
• Lower Bound: The edge connectivity of a connected graph is at least 1. For a disconnected graph, the edge connectivity is 0.
• Upper Bound: The edge connectivity of a graph is at most n-1, where n is the number of vertices in the graph.
• Graph Completeness: For complete graphs, the edge connectivity is n-1, as removing n-1 edges disconnects the graph.

Star Graph Edge Connectivity

The star graph has a central vertex connected to all other vertices. Removing any one of the edges connecting the central vertex to an outer vertex will disconnect that vertex from the rest. Therefore, the edge connectivity of a star graph is 1.

The following image displays a star graph with edge connectivity of 1 −

Complete Graph Edge Connectivity

A complete graph has an edge between every pair of vertices. Since the graph is fully connected, removing any edge still keeps the graph connected. Therefore, the edge connectivity of a complete graph with 5 vertices is 4 (one less than the total number of vertices).

The following image displays a complete graph with edge connectivity of 4 −

Types of Edge Connectivity

There are two main types of edge connectivity, they are −

• Global Edge Connectivity
• Local Edge Connectivity

Global Edge Connectivity

Global edge connectivity refers to the overall connectivity of the graph. It is defined as the smallest number of edges that must be removed to disconnect the entire graph into at least two disconnected subgraphs.

In other words, it measures how strong the graph is against the removal of edges. A graph with higher global edge connectivity is more resistant to disconnection.

Example of Global Edge Connectivity

Consider a complete graph with four vertices, labeled A, B, C, and D, where every vertex is connected to every other vertex. In this case, removing any three edges from the graph would disconnect the graph, as the remaining edges would not keep the graph connected.

Therefore, the global edge connectivity of this graph is 3, as removing three edges is the minimum number required to disconnect the entire graph.

• Graph: A - B - C - D (with edges between every pair of vertices)
• Global Connectivity: 3 (removing three edges disconnects the graph)

Local Edge Connectivity

Local edge connectivity refers to the connectivity around a particular vertex in the graph. It is defined as the minimum number of edges that must be removed to disconnect that specific vertex from the rest of the graph. Local edge connectivity measures how critical a vertex is to the overall connectivity of the graph.

Example of Local Edge Connectivity

Consider a star graph with five vertices: one central vertex (A) and four outer vertices (B, C, D, E) connected only to the central vertex.

In this case, removing any one of the edges connecting the central vertex (A) to an outer vertex (B, C, D, or E) does not disconnect the central vertex from the rest of the graph. However, removing all edges connecting the central vertex will disconnect the graph.

Therefore, the local edge connectivity for any outer vertex (B, C, D, or E) is 1, as removing just one edge is enough to disconnect that specific outer vertex from the graph.

• Graph: A (center) connected to B, C, D, E (outer vertices)
• Local Connectivity (for any outer vertex): 1 (removing one edge disconnects the outer vertex)

Applications of Edge Connectivity

Edge connectivity plays an important role in many applications, such as −

• Network Reliability: In communication networks, edge connectivity shows how strong the network is against edge failures. A higher edge connectivity means the network is less likely to be disconnected if some edges fail.
• Graph Partitioning: Edge connectivity helps in dividing large networks or graphs into smaller subgraphs, ensuring that the partition does not disconnect parts of the graph.
• Communication Networks: Edge connectivity is important for designing communication networks, where the goal is to ensure the network remains connected even after some edges fail.
• Fault Tolerance: A graph with high edge connectivity can handle edge failures without losing its connectivity, making the network more reliable and efficient.

For instance, in a transportation network, each edge represents a transportation link. If one link fails, the connectivity of the network is impacted. Higher edge connectivity ensures that the failure of one link does not cause a breakdown in the transportation system.

Theorems in Edge Connectivity

Several important theorems related to edge connectivity help us understand its relationship with vertex connectivity and provide useful methods for analyzing graphs. Two major theorems are −

• Knig's Theorem
• Menger's Theorem

Knig's Theorem

Knig's Theorem states that in a connected graph, the edge connectivity is equal to the vertex connectivity of the graph.

Edge connectivity refers to the minimum number of edges that need to be removed to disconnect the graph, while vertex connectivity refers to the minimum number of vertices that must be removed to achieve the same result. According to this theorem, these two quantities are equal in a connected graph.

Example of Knig's Theorem

Consider a simple graph where vertices A, B, and C are connected by edges. If we remove one edge (say, the edge between A and B), the graph will become disconnected, as B will no longer be connected to C.

In this case, removing one edge results in disconnection, so the edge connectivity is 1. Similarly, if we remove the vertex B, the graph will also become disconnected. Thus, the edge connectivity (1) is equal to the vertex connectivity (1).

Menger's Theorem

Menger's Theorem provides a more general view of edge connectivity, stating that the minimum number of edges that need to be removed to disconnect a graph is equal to the maximum number of edge-disjoint paths between any two vertices in the graph.

Edge-disjoint paths are paths that do not share any common edge except for the start and end vertices.

Example of Menger's Theorem

Consider a graph where there are two edge-disjoint paths between two vertices, A and B. These paths do not share any common edge except for A and B.

According to Menger's Theorem, at least two edges must be removed to disconnect A from B. This is because the number of edge-disjoint paths between A and B (2) directly corresponds to the minimum number of edges required to disconnect the two vertices.

Removing fewer than two edges will not disconnect A and B, as there will still be paths connecting them.