Question

Draw a simple undirected graph G that has 12 vertices, 18 edges, and 3 connected components. Why would it be impossible to draw G with 3 connected components if G had 66 edges?

Short answer

A graph with 12 vertices, 18 edges, and 3 connected components can be drawn, but a graph with 66 edges would form only 1 fully connected component.

Step-by-step solution

- Understand the Problem

The goal is to draw an undirected graph with 12 vertices (V = 12), 18 edges (E = 18), and 3 connected components.

- Definitions and Basic Concepts

An undirected graph is a set of vertices connected by edges where the edges do not have a directional aspect. A connected component of a graph is a maximal subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices.

- Draw the Graph

Draw 3 separate groups of vertices, ensuring all together they total 12 vertices. Connect the vertices within each group so that the total number of edges across all groups is 18.

- Verify the Number of Edges and Components

1. Divide 12 vertices among 3 components, which gives an average of 4 vertices per component.2. To have a connected graph with 4 vertices in each component, each component must have a minimum of 3 edges (as in a tree structure).3. The remaining edges can be added arbitrarily to ensure a total of 18 edges across the graph.

- Explanation of the Impossibility for 66 Edges

For 12 vertices, the maximum number of edges is given by \(\frac{12 \times (12 - 1)}{2} = 66 \). Having 66 edges means that all possible pairs of vertices are connected, implying there is only one connected component. Therefore, it is impossible to have 3 connected components with 66 edges because the graph would be fully connected.

Key concepts

undirected graph

In graph theory, an undirected graph is a fundamental concept. It consists of a set of vertices and edges where each edge connects a pair of vertices. Unlike directed graphs, edges in undirected graphs do not have a direction. This means that if there is an edge between vertex A and vertex B, you can traverse it from A to B or from B to A without any additional indication.

Key Points:

• Edges are unordered pairs of vertices.
• Traversal can occur in any direction along an edge.
• Suitable for modeling bidirectional relationships.

For example, in a network of friends, if two people (vertices) are friends (an edge), the friendship connection goes both ways. Practical applications include social networks, communication networks, and circuit designs.

connected components

Connected components are another crucial concept in graph theory. A connected component of an undirected graph is a maximal subgraph where any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph. Essentially, it's like an isolated cluster in the graph.

Understanding Connected Components:

• A connected component can be as small as a single vertex with no edges or as large as encompassing multiple vertices and edges.
• Each connected component is disjoint, meaning no vertex belongs to more than one component.
• The decomposition of a graph into connected components can simplify many graph algorithms.

Consider an undirected graph with 12 vertices and 3 connected components. This means the vertices are grouped in such a way that each group (component) is internally connected, but no connections exist between the groups. This concept is useful in network analysis, where discovering isolated sub-networks can provide insights into the structure and behavior of the network.

maximum number of edges

The maximum number of edges in an undirected graph is a critical parameter when analyzing graph connectivity. For a graph with V vertices, the maximum number of edges (E) it can have is given by the formula \[\frac{V \times (V - 1)}{2}\]. This formula assumes that every possible pair of vertices in the graph is connected by a unique edge.

Important Insights:

• This formula emerges from the concept of combinations, specifically choosing 2 vertices out of V.
• When a graph has the maximum number of edges, it is referred to as a complete graph.
• The presence of the maximum number of edges implies that the graph is fully connected, forming a single connected component.

In the context of the provided exercise, for a graph with 12 vertices, the maximum number of edges is calculated as \[\frac{12 \times (12 - 1)}{2} = 66\]. If the graph had 66 edges, all vertices would be interconnected, indicating just one connected component, contrary to the requirement of 3 connected components. This demonstrates the impossibility of having 3 connected components with 66 edges.