Let G = ( V , E ) G = (V, E) G = ( V , E ) Be A Connected Graph Where Every Vertex Has Degree Either 1 1 1 Or 2 2 2 . Then G G G Must Be Either A Single Path Or A Single Cycle.

Mar 9, 2025 by ADMIN 177 views

**The Uniqueness of Graphs with Vertices of Degree 1 or 2**

Introduction

In graph theory, a connected graph is a graph in which there is a path between every pair of vertices. A graph with vertices of degree 1 or 2 is a graph where every vertex has either one or two edges connected to it. In this article, we will explore the properties of such graphs and prove that they must be either a single path or a single cycle.

Theorem Statement

Let $G = (V, E)$ be a connected graph where every vertex has degree either $1$ or $2$ . Then $G$ must be either a single path or a single cycle.

Proof

To prove this theorem, we will use a combination of mathematical induction and graph theory concepts.

Step 1: Base Case

Let's start by considering a graph with only one vertex. In this case, the graph is a single vertex, which is both a path and a cycle. Therefore, the base case is true.

Step 2: Inductive Hypothesis

Assume that the theorem is true for all graphs with $n$ vertices, where $n \geq 1$ . That is, assume that if every vertex in a graph with $n$ vertices has degree either $1$ or $2$ , then the graph must be either a single path or a single cycle.

Step 3: Inductive Step

Now, let's consider a graph $G$ with $n+1$ vertices, where every vertex has degree either $1$ or $2$ . We need to show that $G$ must be either a single path or a single cycle.

Let $v$ be a vertex in $G$ with degree $2$ . Since $v$ has degree $2$ , there are two edges connected to $v$ . Let $u$ and $w$ be the vertices connected to $v$ by these two edges.

If $u$ and $w$ are not connected by an edge, then $G$ is a single path with $n+1$ vertices. Therefore, we can assume that $u$ and $w$ are connected by an edge.

Now, consider the subgraph $H$ induced by the vertices $u$ , $v$ , and $w$ . Since $u$ and $w$ are connected by an edge, $H$ is a cycle with three vertices. Therefore, $H$ is a single cycle.

Since $v$ has degree $2$ , there are no other edges connected to $v$ in $G$ . Therefore, $G$ is the union of the single cycle $H$ and the remaining vertices in $G$ . Since $G$ is connected, the remaining vertices in $G$ must be connected to $H$ by an edge.

Let $x$ be a vertex in the remaining vertices in $G$ . Since $x$ has degree either $1$ or $2$ , there is an edge connected to $x$ . If the edge connected to $x$ is connected to a vertex in $H$ , then $G$ is a single cycle. Otherwise, the edge connected to $x$ is connected to a vertex $y$ not in $H$ . Since $y$ has degree either $1$ or $2$ , there is an edge connected to $y$ . If the edge connected to $y$ is connected to a vertex in $H$ , then $G$ is a single cycle. Otherwise, the edge connected to $y$ is connected to a vertex $z$ not in $H$ . Since $z$ has degree either $1$ or $2$ , there is an edge connected to $z$ . If the edge connected to $z$ is connected to a vertex in $H$ , then $G$ is a single cycle. Otherwise, we can continue this process until we reach a vertex in $H$ .

Therefore, we can conclude that $G$ is the union of the single cycle $H$ and the remaining vertices in $G$ . Since $G$ is connected, the remaining vertices in $G$ must be connected to $H$ by an edge. Therefore, $G$ is a single cycle.

Conclusion

We have shown that if every vertex in a graph has degree either $1$ or $2$ , then the graph must be either a single path or a single cycle. This completes the proof of the theorem.

Corollary

As a corollary to this theorem, we can conclude that a graph with vertices of degree $1$ or $2$ cannot have any vertices with degree greater than $2$ . This is because if a vertex has degree greater than $2$ , then it cannot be part of a single path or single cycle.

Example

Let's consider an example of a graph with vertices of degree $1$ or $2$ . Suppose we have a graph with $5$ vertices, where every vertex has degree either $1$ or $2$ . We can draw this graph as follows:

  A
 / \
B---C
 \ /
  D

In this graph, every vertex has degree either $1$ or $2$ . Therefore, by the theorem, this graph must be either a single path or a single cycle. In this case, the graph is a single cycle.

Conclusion

Q: What is the significance of the theorem that a graph with vertices of degree 1 or 2 must be either a single path or a single cycle?

A: The theorem has significant implications for graph theory and has many applications in computer science and other fields. For example, it can be used to analyze the structure of networks and to identify patterns in data.

Q: Can a graph with vertices of degree 1 or 2 have any vertices with degree greater than 2?

A: No, a graph with vertices of degree 1 or 2 cannot have any vertices with degree greater than 2. This is because if a vertex has degree greater than 2, then it cannot be part of a single path or single cycle.

Q: What is an example of a graph with vertices of degree 1 or 2 that is not a single path or single cycle?

A: Actually, the theorem states that a graph with vertices of degree 1 or 2 must be either a single path or a single cycle. Therefore, there are no examples of graphs with vertices of degree 1 or 2 that are not a single path or single cycle.

Q: Can a graph with vertices of degree 1 or 2 have any isolated vertices?

A: No, a graph with vertices of degree 1 or 2 cannot have any isolated vertices. This is because if a vertex is isolated, then it has degree 0, which is not allowed.

Q: How can the theorem be applied to real-world problems?

A: The theorem can be applied to real-world problems in many ways. For example, it can be used to analyze the structure of social networks, to identify patterns in data, and to optimize the design of networks.

Q: What are some common applications of the theorem?

A: Some common applications of the theorem include:

Network analysis: The theorem can be used to analyze the structure of networks and to identify patterns in data.
Data mining: The theorem can be used to identify patterns in data and to optimize the design of data mining algorithms.
Computer science: The theorem has many applications in computer science, including the design of algorithms and the analysis of networks.

Q: Can the theorem be extended to graphs with vertices of degree 3 or more?

A: No, the theorem cannot be extended to graphs with vertices of degree 3 or more. This is because the theorem relies on the fact that every vertex has degree 1 or 2, which is not true for graphs with vertices of degree 3 or more.

Q: What are some open problems related to the theorem?

A: Some open problems related to the theorem include:

Can the theorem be extended to graphs with vertices of degree 3 or more?
Can the theorem be applied to more general types of graphs, such as directed graphs or weighted graphs?
Can the theorem be used to analyze the structure of more complex networks, such as networks with multiple layers or networks with non-linear relationships?

Conclusion

In this article, we have answered some common questions related to the theorem that a graph with vertices of degree 1 or 2 must be either a single path or a single cycle. We have also discussed some of the applications and implications of the theorem, as well as some open problems related to the theorem.