Edge Set Cardinality - 4.3.2 | 4. Prof. Ashish Choudhury | Discrete Mathematics - Vol 3
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Interactive Audio Lesson

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Understanding Graphs

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Teacher
Teacher

Today, we'll discuss graphs, which consist of vertices and edges. Can anyone tell me what 'vertex connectivity' refers to in a graph?

Student 1
Student 1

Is it about how many vertices need to be removed to disconnect the graph?

Teacher
Teacher

Exactly! So, vertex connectivity is the minimum number of vertices whose removal increases the number of connected components. We also have edge connectivity. Can anyone tell me what that is?

Student 2
Student 2

It's the minimum number of edges that need to be removed to disconnect the graph.

Teacher
Teacher

Right! And to remember these two concepts, we can use the acronym V.E.C. V for Vertex connectivity, E for Edge connectivity, and C for connectivity itself. Let's move on to the relationship between vertex and edge connectivity.

Hierarchy of Connectivity

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Teacher
Teacher

Recall the hierarchy we mentioned earlier: vertex connectivity should be less than or equal to edge connectivity, and edge connectivity in turn should be less than or equal to the minimum degree. Why do we think that's the case?

Student 3
Student 3

Because if you disconnect all vertices, then you must also disconnect the edges.

Teacher
Teacher

Exactly! This is a key concept in understanding how graph structures work. Now, let's use this knowledge to construct a graph. If l = 3, m = 4, and n = 5, how can we ensure these properties?

Student 4
Student 4

We can use two complete graphs with n+1 nodes, right?

Teacher
Teacher

Correct! Each complete graph ensures a high minimum degree. Let's illustrate how we can pick vertices and add special edges to satisfy our conditions.

Edge Set Cardinality Calculation

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Teacher
Teacher

Now, let's talk about edge set cardinality. What happens when you remove a vertex from a graph? How does that impact edge count?

Student 1
Student 1

The edges connected to that vertex disappear too, so the edge count decreases by the vertex's degree.

Teacher
Teacher

Yes! So, if deleting vertex v_i leaves us with a certain number of edges, we can set up equations to find edge set cardinality. If we have six vertices and various edges, how can we calculate the total?

Student 2
Student 2

We need to consider the degree of each vertex after deletion and sum those to find the total edges.

Teacher
Teacher

Great! By using the summation of degrees and applying the handshaking theorem, we can derive our edge set cardinality equation. This is crucial for a deeper understanding of graph theory.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section discusses the cardinality of the edge set in graphs and the relationships between various connectivity measures like vertex connectivity and edge connectivity.

Standard

In this section, we explore the properties of edge sets in graphs, focusing on how to construct simple graphs that satisfy specific vertex and edge connectivity conditions. Key relationships and theorems are also examined to derive cardinalities and connectivity properties.

Detailed

Edge Set Cardinality

This section delves into the important aspects of edge set cardinality in graph theory. We begin with an exploration of a scenario where we define a simple graph given three positive integers: vertex connectivity (l), edge connectivity (m), and minimum degree (n). The relationships between these parameters are significant since they define the structural soundness of graphs. The fundamental relationships are:
- Vertex connectivity (κ) ≤ Edge connectivity (λ) ≤ Minimum degree (δ).

To construct a graph that adheres to these specified properties, we focus on creating two complete graphs, denoted as K_(n+1), each with n+1 nodes. The minimum degree of the final graph must be at least n, which can be ensured by utilizing these complete graphs. By selecting specific vertices from each graph and adding edges between them strategically, we can achieve our desired vertex connectivity (l) and edge connectivity (m).

Further, the section discusses calculating the cardinality of the edge set of a graph based on the removal of vertices and how this affects the count of edges. Using logical reasoning and properties of graphs, we arrive at the ability to determine edge set cardinality from the degrees of vertices after removal.

The section concludes with practical examples that illustrate the construction and derivation of edge set cardinality in context, allowing students to grasp the foundational principles underlying these concepts.

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Audio Book

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Understanding Edge Set Cardinality

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In question number 2 you are given an unknown simple graph G, the graph G is not known to you known to you in the sense that you are just given that it has 6 vertices but exact cardinality of edge set is not given. But it is given to you that your graph G is such that if you delete the vertex v from the graph then you are left with 7 edges. If you delete vertex v from the graph you are left with 6 edges and so on.

Detailed Explanation

In this chunk, we explore the idea of edge set cardinality in a graph. Given that a graph G has 6 vertices, we're interested in identifying how many edges (the edge set cardinality) are present initially. We have information about the number of edges remaining after removing specific vertices from the graph G. Essentially, when a vertex is removed from a graph, the edges connected to that vertex are also removed, which reduces the total number of edges in the graph. For instance, removing vertex v1 leaves 7 edges, meaning that vertex v1 was incident to a certain number of edges which we will determine.

Examples & Analogies

To think about this, imagine a group of six friends connected in various ways – some are close friends and share mutual connections (edges). If one friend (a vertex) leaves the group, the connections they had (edges) are disrupted. By looking at how many connections remain when each friend leaves, we can deduce how well-connected our group is, akin to finding the edge set cardinality.

Finding Edge Cardinality

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So, if that is the case the question asks you to find out the cardinality of the edge set of the original graph. Again do not try to do a brute force and try all possible graphs in your mind and then hit upon the answer because that will take enormous amount of time.

Detailed Explanation

This chunk invites us to think critically and analytically rather than through brute-force trial and error. The cardinality of the edge set can be derived using known properties of graph theory. We won't have to list every possible graph to find the edge count; instead, we can use the relationships we’ve established while looking at edges left after removing vertices. This approach saves time and increases efficiency by relying on mathematical logic and insights from the structure of the graph.

Examples & Analogies

Imagine you're trying to guess how many cards are in a deck without counting them directly. Instead, you know that when you remove a known group of cards, you can determine how many were there originally based on what’s left. This is similar to how we assess the edges in a graph.

Applying the Degree of Vertices

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If you have the vertex v here and if you remove the vertex v from the graph the cardinality of the edge set gets decremented by the degree of v because the deletion of the vertex v will delete how many edges from the graph? All the edges which are incident with the vertex v namely degree of v number of edges from my edge set will be removed.

Detailed Explanation

Here, we discuss the impact of removing a vertex on the edge set of a graph. The degree of a vertex indicates how many edges are connected to that vertex. When a vertex is removed, we lose those edges connected to it, thereby reducing the total number of edges in the graph. So, if we have several equations based on different vertices and their respective degrees, we can deduce the original total edge cardinality.

Examples & Analogies

Think of a network of water pipes where each junction (vertex) has a certain number of pipes (edges) connected. If you remove a junction, all the pipes connected to it are also removed, leading to less flow in the entire network. This analogy helps to visualize the concept of edge removals based on vertex degrees.

Summing Up the Equations

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Now if I sum all these 6 equations I get that 6 times the cardinality of E minus the summation of the degree of 6 vertices in the graph is 36.

Detailed Explanation

We gather information from the previously established relationships about vertex removals. By summing all the equations derived from removing a vertex and utilizing their degrees, we can find a single equation that relates directly to the original edge count. This consolidation of information allows us to simplify the problem and focus on just one key equation that leads us to the answer.

Examples & Analogies

If you were to tally the number of friends removed from a gathering and how many friendships they represented, summing those counts can help you figure out just how connected the entire group was before anyone left. It's a generalization of many smaller observations into one clear picture.

Using the Handshaking Theorem

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And now I can apply the handshaking theorem which says that if you take the summation of the degrees of all the vertices in your original graph it is the same as twice the number of edges.

Detailed Explanation

The Handshaking Theorem forms a pivotal concept in understanding the relationships between the vertices' degrees and edges in a graph. It tells us that if we sum the degrees of all the vertices in a graph, we will end up with twice the number of edges, given that each edge connects two vertices and counts towards both of their degrees. This relationship is essential for solving for edge cardinality in our earlier equations.

Examples & Analogies

Picture a social gathering: each handshake between two people creates a connection. If we list every person and count how many people they've shaken hands with, we actually count each handshake between two friends twice. Thus, summing all individual counts gives us double the actual number of handshakes, just as the theorem suggests for edges in a graph.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Connectivity: A crucial characteristic, determining how many vertices or edges can be removed before a graph becomes disconnected.

  • Edge Set: The collection of edges in a graph, whose cardinality is essential for understanding connectivity.

  • Graph Construction: The process of creating a graph based on specific connectivity requirements.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example 1: Construct a graph with vertex connectivity 3, edge connectivity 4, and minimum degree 5 using two complete graphs.

  • Example 2: Determine the edge set cardinality of a graph after removing vertices and relating it to the vertex degrees.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • To cut the graph in two, just take a few, vertices or edges, that's the clue.

📖 Fascinating Stories

  • Once in a land of graphs, two complete castles stood tall. To connect them, a few knights had to be removed, ensuring the castles remained strong even as some edges fell away.

🧠 Other Memory Gems

  • Remember 'V.E.C.' for Vertex, Edge, Connectivity - understanding the flow between them.

🎯 Super Acronyms

CEM (Connectivity, Edge, Minimum) helps you remember the order of importance in graphs.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Vertex Connectivity

    Definition:

    The minimum number of vertices that must be removed to disconnect a graph.

  • Term: Edge Connectivity

    Definition:

    The minimum number of edges that must be removed to disconnect a graph.

  • Term: Minimum Degree

    Definition:

    The smallest degree among all the vertices in a graph.

  • Term: Cardinality

    Definition:

    The number of elements in a set, often used in the context of counting edges in graph theory.