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.

The Uniqueness of Graphs with Vertices of Degree 1 or 2

In the realm of graph theory, understanding the properties of connected graphs is crucial for solving various problems. A connected graph is a graph in which there is a path between every pair of vertices. In this article, we will explore the properties of connected graphs where every vertex has a degree of either 1 or 2. We will prove that such graphs 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.

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

Base Case

Let's consider a graph with only two vertices, $u$ and $v$. Since every vertex has a degree of either 1 or 2, the only possible graph is a single edge between $u$ and $v$. This graph is both a single path and a single cycle, satisfying the theorem statement.

Inductive Step

Assume that the theorem holds for all connected graphs with $n$ vertices, where $n \geq 2$. We need to show that the theorem holds for a connected graph with $n+1$ vertices.

Let $G = (V, E)$ be a connected graph with $n+1$ vertices, where every vertex has degree either 1 or 2. We will show that $G$ must be either a single path or a single cycle.

Case 1: A Vertex with Degree 1

Let $v$ be a vertex in $G$ with degree 1. Since $G$ is connected, there must be a vertex $u$ adjacent to $v$. Let $e$ be the edge between $u$ and $v$. We can remove $e$ from $G$ to obtain a connected graph $G'$ with $n$ vertices.

By the inductive hypothesis, $G'$ must be either a single path or a single cycle. Since $e$ is a single edge, adding $e$ back to $G'$ will result in a single path or a single cycle, satisfying the theorem statement.

Case 2: A Vertex with Degree 2

Let $v$ be a vertex in $G$ with degree 2. Since $G$ is connected, there must be two vertices $u_1$ and $u_2$ adjacent to $v$. Let $e_1$ and $e_2$ be the edges between $u_1$ and $v$, and $u_2$ and $v$, respectively.

We can remove $e_1$ and $e_2$ from $G$ to obtain a connected graph $G'$ with $n-1$ vertices. By the inductive hypothesis, $G'$ must be either a single path or a single cycle.

Since $e_1$ and $e_2$ are two edges, adding them back to $G'$ will result in a single cycle, satisfying the theorem statement.

Conclusion

We have shown that if $G = (V, E)$ is a connected graph where every vertex has degree either 1 or 2, then $G$ must be either a single path or a single cycle. This completes the proof of the theorem.

The theorem has several interesting implications. For example, it shows that a connected graph with vertices of degree 1 or 2 cannot have any cycles of length greater than 2. This is because any cycle of length greater than 2 would require a vertex with degree greater than 2.

The theorem also has applications in computer science, where it can be used to analyze the structure of graphs representing networks or databases.

In this article, we have proved that a connected graph where every vertex has degree either 1 or 2 must be either a single path or a single cycle. This theorem has several interesting implications and applications in graph theory and computer science.

• [1] Bondy, J. A., & Murty, U. S. R. (2008). Graph theory. Springer.
• [2] Diestel, R. (2017). Graph theory. Springer.

Note: The references provided are for general graph theory texts and are not specific to the theorem proved in this article.
Q&A: Understanding Graphs with Vertices of Degree 1 or 2

In our previous article, we proved that a connected graph where every vertex has degree either 1 or 2 must be either a single path or a single cycle. In this article, we will answer some frequently asked questions about this theorem and provide additional insights into the properties of such graphs.

Q: What is the significance of a graph having vertices of degree 1 or 2?

A: A graph with vertices of degree 1 or 2 is a special type of graph that has some unique properties. For example, such graphs cannot have any cycles of length greater than 2. This is because any cycle of length greater than 2 would require a vertex with degree greater than 2.

Q: Can a graph with vertices of degree 1 or 2 have multiple connected components?

A: No, a graph with vertices of degree 1 or 2 cannot have multiple connected components. This is because if a graph has multiple connected components, then there must be a vertex with degree greater than 2, which is not allowed.

Q: How can we visualize a graph with vertices of degree 1 or 2?

A: A graph with vertices of degree 1 or 2 can be visualized as a single path or a single cycle. For example, if we have a graph with 5 vertices, where every vertex has degree 1 or 2, then the graph must be either a single path of length 5 or a single cycle of length 5.

Q: Can a graph with vertices of degree 1 or 2 have any vertices with degree 0?

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

Q: How can we use this theorem in real-world applications?

A: This theorem has several real-world applications, such as:

• Network analysis: This theorem can be used to analyze the structure of networks, such as social networks or communication networks.
• Database design: This theorem can be used to design databases that have a specific structure, such as a single path or a single cycle.
• Computer science: This theorem can be used to analyze the structure of algorithms and data structures.

Q: Can a graph with vertices of degree 1 or 2 have any self-loops?

A: No, a graph with vertices of degree 1 or 2 cannot have any self-loops. This is because a self-loop would require a vertex with degree greater than 2, which is not allowed.

In this article, we have answered some frequently asked questions about the theorem that a connected graph where every vertex has degree either 1 or 2 must be either a single path or a single cycle. We have also provided additional insights into the properties of such graphs and their real-world applications.

• [1] Bondy, J. A., & Murty, U. S. R. (2008). Graph theory. Springer.
• [2] Diestel, R. (2017). Graph theory. Springer.

Note: The references provided are for general graph theory texts and are not specific to the theorem proved in this article.