4.1.4. Tutorial 9: Part I

In this question, you are given 3 positive numbers l, m, n, not be positive, non-negative integers. So, l, m, n such that l is less than equal to m and m is less than equal to n.

In this introduction, we are discussing a problem involving three variables: l, m, and n which represent different connectivity metrics of a graph. Specifically, l is the vertex connectivity, m is the edge connectivity, and n is the minimum degree of the graph. The problem states constraints between these three variables, namely, l cannot exceed m, and m cannot exceed n. This relationship sets the stage for constructing a graph with specific properties based on these metrics.

Think of a social network where people are nodes. The relationships between them are the edges. The values l, m, and n represent the connectivity of this network: l could represent how many friends you need to remove before the network becomes disconnected (vertex connectivity), m could represent how many direct connections need to be severed to make a person isolated (edge connectivity), and n could represent the minimum number of friends any person has in the network.

So, here is how we can construct a graph. Since we need the minimum degree in the graph to be n, to ensure that my resultant graph; my final graph has the minimum degree n, I take 2 copies of a complete graph with n + 1 nodes.

To fulfill the requirement for minimum degree n in our graph, we start by creating two complete graphs, each containing n + 1 nodes. A complete graph is one in which every pair of distinct vertices is connected by a unique edge. By doing this, we ensure that every vertex in each complete graph has degree n, satisfying the minimum degree requirement. Therefore, we are building a robust structure that allows us to adjust the connectivity metrics in subsequent steps.

Imagine building a strong network of roads connecting several towns (n towns plus the central town makes n + 1). By ensuring every town is directly connected to every other town, we are preparing for a scenario where any town can be reached from any other without blocked roads. This mirrors setting up an initial strong social network before we make specific adjustments.

Now I have to take care of my vertex connectivity and edge connectivity. So, how do I do that? I randomly pick l nodes and m nodes from the two copies.

After establishing two complete graphs, the next step is to ensure that the overall graph meets the required vertex and edge connectivity. To achieve this, we will select l nodes from the first copy of the complete graph and m nodes from the second copy. This action will allow us to introduce connections between these selected nodes that help define the desired connectivity metrics of the overall graph.

Consider a sports team: to ensure strong team performance, you may pick a specific number of players from each squad to ensure that teams are well-connected (like ensuring there are strong connections between offensive and defensive roles). By carefully selecting players from both teams, you can set the stage for better collaboration and coordination.

I will add edges between the l nodes which I have picked in the first copy and m nodes which I have picked in the second copy in such a way that it ensures that I add basically m edges.

Next, we need to establish the vertex and edge connectivity through the addition of special edges. We will add exactly m edges connecting the l nodes taken from the first graph to the m nodes from the second graph. The configuration of these edges is critical as they define the connectivity properties of the new graph formed from the two copies.

Think of connecting different sections of a city—adding roads (edges) between different neighborhoods (nodes). By planning these connections carefully, you create a seamless transport network, analogous to ensuring connectivity in our graph.

So, if l = m or if l would have been equal to m what I would have done is I would have picked l edges in the first copy and m edges in the second copy and just add distinct edges.

This section discusses the special case where the number of selected nodes l equals m. If that happens, we can connect each l node from the first graph to a corresponding m node in the second graph without repetitions, ensuring distinct connections. This connection strategy guarantees we meet the connectivity specifications evenly.

Imagine pairing students from two different classes for a project. If you have the same number of students from each class, it's easy to assign one student from one class to work with one student from the other class without overlaps, ensuring equal participation.

Now we can see that the way I have given these special edges it is ensured that my vertex connectivity is l why vertex connectivity is l? Because if I delete the l vertices which I have picked or which are the endpoints of the special edges from the first copy of the K graph sub graph then my entire graph gets disconnected.

Having constructed the connections between the selected nodes, we can verify that the vertex connectivity of the graph is now indeed l. If we were to remove the l nodes that form the endpoints of our special edges, the graph would become disconnected, fulfilling the vertex connectivity requirement.

Think of a campus filled with important departments. If we were to remove the department heads from those connections, the entire administrative structure might fall apart. The significance of those connections is similar to the l nodes we selected.

And it is easy to see that the edge connectivity is m because the m edges which I have added across the 2 copies of K that constitute the edge cut.

In conclusion, we determine the edge connectivity of the newly formed graph. Since we added m edges connecting our selected nodes, removing these edges will also cause the graph to become disconnected, thus verifying that the edge connectivity is indeed m.

Finally, consider an intricate network of roads. If certain key roads are removed, the connections between various regions may be severed, verifying their importance in maintaining traffic flow, comparable to edge connectivity in our graph.

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