Number of Connected Components in a Graph

In graph theory, a connected component of an undirected graph is a subgraph in which there is a path between every pair of vertices. In other words, all vertices in a connected component are reachable from each other.

Minimum Number of Connected Components

To determine the minimum number of connected components in a graph with 20 vertices and 12 edges, we need to consider the worst-case scenario where the graph is disconnected as much as possible.

Maximum Number of Disconnected Components

To find the maximum number of disconnected components in a graph, we can consider each edge as a potential separator. In the worst-case scenario, each edge can separate two distinct connected components, resulting in a disconnected graph.

Calculating the Maximum Number of Disconnected Components

To calculate the maximum number of disconnected components, we subtract the number of edges from the number of vertices and add 1. This is because each edge can potentially separate two connected components, and the remaining vertices form a separate component.

Maximum Number of Disconnected Components = Number of Vertices - Number of Edges + 1
= 20 - 12 + 1
= 9

Minimum Number of Connected Components

The minimum number of connected components is the complement of the maximum number of disconnected components.

Minimum Number of Connected Components = Total Number of Components - Maximum Number of Disconnected Components
= 1 - 9
= -8

However, the number of connected components cannot be negative. Therefore, we take the absolute value of the result.

Minimum Number of Connected Components = | -8 |
= 8

Therefore, the minimum number of connected components in a simple graph with 20 vertices and 12 edges is 8.