4.1.2. International Institute of Information Technology - Bangalore

Hello everyone, welcome to the first part of tutorial 9 so, let us start with question number 1. So, 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 set the stage for solving a problem involving the construction of a graph based on specific criteria. We are given three key parameters: l, m, and n, which represent different connectivity and degree conditions for a graph. The relationships among these parameters are critical for finding a valid graph structure.

Think of planning a network of friends. Here, l is the minimum number of friendships each person must have (vertex connectivity), m is how interconnected the friendships are across groups (edge connectivity), and n is the minimum number of friends someone should have (minimum degree).

And what we want here is a simple graph where the vertex connectivity is l, edge connectivity is m and minimum degree is n. So, remember the relationship between the vertex connectivity edge connectivity, and the minimum degree is that: vertex connectivity is less than equal to edge connectivity and edge connectivity is less than equal to the minimum degree in the graph.

This chunk explains the specific requirements for the graph we want to construct. The relationships among l, m, and n provide constraints that must be respected in any graph that fulfills the requirements. The inequalities highlight the hierarchy of connectivity and degree constraints that guide our graph construction.

Imagine a city where roads connect various neighborhoods. The vertex connectivity (l) is the minimum number of roads required to keep a neighborhood connected, edge connectivity (m) is the minimum number of overall intersections that must remain for good connectivity, and the minimum degree (n) is how many roads each neighborhood should ideally have.

Basically in this question we are asking you to give the construction of one simple graph which satisfies the inequality with respect to the l, m, n values that are given to you. 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 construct the desired graph, we first ensure it meets the minimum degree requirement by creating 2 copies of a complete graph with n + 1 nodes. This foundational step guarantees that we start with a robust structure capable of supporting the defined connectivity and degree parameters.

Suppose you’re constructing a community center. By starting with two large interconnecting buildings (complete graphs), you ensure that each building (node) can be well connected to a sufficient number of recreational rooms (edges) which allows flexibility in how activity areas are designed.

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. So, l nodes I pick from the first copy and m nodes I pick from the second copy.

This chunk focuses on selecting specific nodes from the two copies of the complete graph to establish the necessary vertex and edge connectivity. By picking l nodes from one graph and m from another, we form the basis for additional connections that will satisfy our structural requirements.

Imagine picking students (l from one class and m from another) to represent your school in a sports team. One class has a certain number of players (l), and the other class a different number (m). The selected students will determine how strong the team is and how it connects with other teams during a tournament.

So, I am going to demonstrate assuming that l = 3 and m = 4. So I have picked 3 nodes arbitrarily from the first copy. And I have picked 4 nodes arbitrarily from the second copy. Now I have to take care; I have to ensure that my vertex connectivity should become l and edge connectivity should become m.

In this section, we demonstrate how specific node selections can be used to determine the newly added edges necessary to achieve the required connectivity. By selecting specific nodes, we can strategically add edges to meet the conditions of connectivity.

Imagine you are forming new team links between students in each selected group to ensure good communication and collaboration. By ensuring that some students are connections (edges) to others between groups, you increase both collaboration (vertex connectivity) and communication efficiency (edge connectivity).

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 first copy m edges in the second copy and just add distinct edges.

This section discusses the scenario when the number of chosen nodes from both graph copies are equal. In such a case, edges can be added distinctly between corresponding nodes to ensure the desired connectivity without overlap, thus maintaining the unique relationships required by the parameters.

Think of arranging a debate between two equal teams. If team sizes are identical, you can match them one to one for arguments. This way, each argument can stand alone without redundancy, allowing for a fair and effective debate structure.

Now you 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 get disconnected.

In this chunk, we verify that our construction fulfills the vertex connectivity requirement. By analyzing what happens when we remove specific nodes (l vertices), we ensure that such a removal leads to disconnection of the graph, thus confirming that our constructed graph adheres strictly to the given connectivity values.

Consider a highway system where certain main exits (vertices) serve as key connectors between regions. If one of those key exits is closed (removed), it creates traffic chaos (disconnection) in that area, showcasing the importance of those exits in maintaining connectivity.

So that ensures that edge connectivity is m and as I argued that since I have taken 2 copies of complete graph with n + 1 nodes which are a sub graph of the entire graph the minimum degree in the graph is at least n so that is the construction.

In conclusion, we have successfully constructed a graph that meets all specified criteria for vertex connectivity, edge connectivity, and minimum degree. We reassert that this construction stands firm under the relationships established by the parameters l, m, and n, completing the task effectively.

Imagine successfully setting up a well-connected transportation network. By ensuring that there are sufficient roads (edges) connecting various towns (vertices), and ensuring a minimum number of links exist, you create a resilient network that effectively meets transportation needs across a region.

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