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Chapter 10: Problem 52

How many subgraphs with at least one vertex does \(W_{3}\) have?

Short Answer

Expert verified
15

Step by step solution

01

Identify the complete graph

\(W_3\) is known as the wheel graph with 4 vertices (3 outer vertices forming a triangle and 1 central vertex connected to all others).
02

Find the total number of subgraphs

For each vertex, it has two choices: either to be included or excluded in a subgraph. Hence, each of the 4 vertices leads to \(2^4\) potential combinations. The total number of subgraphs is \(2^4 = 16\).
03

Subtract the empty subgraph

Among these 16 possible subgraphs, 1 is the empty subgraph (where no vertices are included). Subtract this empty subgraph to get the number of subgraphs with at least one vertex: \(16 - 1 = 15\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wheel Graph
A wheel graph is a special type of graph in graph theory. It is denoted by the symbol \(W_n\), where 'n' represents the number of vertices.
In a wheel graph, there are two main components:
  • Outer vertices that form a cycle (like the rim of a wheel)
  • One central vertex connected to all outer vertices (like the hub of a wheel)
As an example, \(W_3\) has 4 vertices鈥�3 outer vertices forming a triangle and 1 central vertex connected to all the others.
The structure of a wheel graph makes it easy to visualize and work with, especially in problems involving subgraphs and combinatorial enumeration.
Combinatorial Enumeration
Combinatorial enumeration deals with counting the number of ways to arrange or select items from a set. In the problem, we need to count the number of subgraphs of the wheel graph \(W_3\).
When considering a set of vertices, each vertex has two choices: it can either be included in a subgraph or excluded. This leads us to use the combinatorial concept of combinations.
For the wheel graph \(W_3\), which has 4 vertices, each vertex can either be included or not:
This gives us \[ 2^{4} = 16 \] potential combinations of vertices forming subgraphs.
Remember, one of these combinations is the empty subgraph (where no vertices are included).
Subgraph Counting
Counting subgraphs involves determining the number of smaller graphs (subgraphs) that can be formed from a given graph.
In the exercise, we identified the total number of subgraphs of \(W_3\) by calculating \[ 2^{4} = 16 \] combinations based on the inclusion or exclusion of its 4 vertices.
However, we are interested in subgraphs with at least one vertex. To get this, we have to exclude the empty subgraph from our count.
The empty subgraph is one of the 16, so:
We subtract 1 from 16, resulting in \[ 16 - 1 = 15 \] subgraphs that include at least one vertex.
This method helps us accurately determine the number of subgraphs in the problem.

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Most popular questions from this chapter

The thickness of a simple graph \(G\) is the smallest number of planar subgraphs of \(G\) that have \(G\) as their union. $$ \begin{array}{l}{\text { Show that if } G \text { is a connected simple graph with } v \text { ver- }} \\ {\text { tices and } e \text { edges, where } v \geq 3, \text { then the thickness of } G \text { is }} \\ {\text { at least }[e /(3 v-6)] .}\end{array} $$

How many different channels are needed for six stations located at the distances shown in the table, if two stations cannot use the same channel when they are within 150 miles of each other? $$\begin{array}{|c|c|c|c|c|c|}\hline & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline 1 & {-} & {85} & {175} & {200} & {50} & {100} \\ \hline 2 & {85} & {-} & {125} & {175} & {100} & {160} \\ \hline 4 & {200} & {175} & {100} & {-} & {210} & {220} \\ \hline 5 & {50} & {100} & {200} & {210} & {-} & {100} \\\ \hline 6 & {100} & {160} & {250} & {220} & {100} & {-} \\ \hline\end{array}$$

Which of these nonplanar graphs have the property that the removal of any vertex and all edges incident with that vertex produces a planar graph? \(\begin{array}{llll}{\text { a) } K_{5}} & {\text { b) } K_{6}} & {\text { c) } K_{3,3}} & {\text { d) } K_{3,4}}\end{array}\)

Show that if \(G\) is a simple graph with \(n\) vertices, then the union of \(G\) and \(\overline{G}\) is \(K_{n}\) .

The mathematics department has six committees, each meeting once a month. How many different meeting times must be used to ensure that no member is scheduled to attend two meetings at the same time if the committees are \(C_{1}=\\{\text { Arlinghaus, Brand, Zaslavsky }\\}, C_{2}=\\{\text { Brand, }\) Lee, Rosen \(\\}, C_{3}=\\{\text { Arlinghaus, Rosen, Zaslavsky }\\},\) \(C_{4}=\\{\text { Lee, Rosen, Zaslavsky }\\}, \quad C_{5}=\\{\text { Arlinghaus, }\) Brand \(\\},\) and \(C_{6}=\\{\text { Brand, Rosen, Zaslavsky }\\} ?\)
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