Describe an example of a graph where adding a single edge can increase the size of the largest strongly connected component by at least 1,000 nodes. Clearly, you don't need to draw the full graph rather, you should describe it in words.
To describe a graph where adding a single edge can increase the size of the largest strongly connected component by at least 1,000 nodes, let us consider the following example:
1. Start with a graph containing a large strongly connected component (SCC) and several disconnected nodes.
2. The large SCC consists of a cycle of 1,000 nodes, meaning each node is connected to its adjacent nodes, forming a closed loop.
3. Additionally, there are other smaller disconnected nodes that are not part of this SCC.
4. Adding a single edge between any two of the disconnected nodes will create a new connection between them, forming a new SCC.
5. Since the new SCC consists of only two nodes, its size increases by at least 1,000 nodes (as the original SCC had 1,000 nodes).
6. Thus, by adding a single edge, the size of the largest SCC increases by at least 1,000 nodes.
It is important to note that the overall graph structure and connections do not necessarily need to follow a specific pattern, as long as the graph contains a large SCC with disconnected nodes and adding a single edge between disconnected nodes increases the size of the largest SCC by at least 1,000 nodes.