To analyze the problem, let’s break down the given information:

1. Degrees of vertices: In graph $G$, every vertex has either degree 1 or degree 3.
2. Total edges: The graph has 10 edges.
3. Vertices of degree 3: There are exactly 2 vertices of degree 3.

Let’s denote:

• $n_1$: the number of vertices of degree 1.
• $n_3 = 2$: the number of vertices of degree 3.

Step 1: Calculate the total degree

The sum of the degrees of all vertices in a graph is equal to twice the number of edges. Therefore, we can write:

$\text{Total degree} = 2 \times \text{Number of edges} = 2 \times 10 = 20.$

The total degree can also be expressed in terms of the degrees of the vertices:

$\text{Total degree} = 1 \times n_1 + 3 \times n_3 = n_1 + 3 \times 2 = n_1 + 6.$

Setting these two expressions for total degree equal gives:

$n_1 + 6 = 20.$

Solving for $n_1$:

$n_1 = 20 - 6 = 14.$

Step 2: Count the total number of vertices

The total number of vertices $n$ in the graph is the sum of the vertices of degree 1 and degree 3:

$n = n_1 + n_3 = 14 + 2 = 16.$

Step 3: Components and edges

Now, consider the connectivity of the graph. A connected component is formed by vertices connected through edges.

For $k$ components, the number of edges $E$ can be calculated as follows. The minimum number of edges required to connect $n$ vertices in $k$ components is:

$E_{\text{min}} = n - k.$

In our case, since $E = 10$ and $n = 16$:

$E_{\text{min}} = 16 - k.$

Setting this equal to the given number of edges:

$16 - k \leq 10 \implies k \geq 6.$

Conclusion

Since $k$ must be at least 6, having 5 connected components (i.e., $k = 5$) contradicts this condition. Therefore, the graph cannot have 5 connected components.

Summary

We have shown that a graph $G$ with 10 edges, where every vertex has either degree 1 or degree 3, and exactly 2 vertices of degree 3, cannot have 5 connected components.

Would you like further details or have any questions? Here are some related questions for consideration:

1. How would the scenario change if the number of edges were increased?
2. What would happen if there were 3 vertices of degree 3 instead of 2?
3. Can you describe the structure of a possible graph with the given conditions?
4. What are the implications of having vertices with higher degrees?
5. How does the connectivity of a graph affect its components?

Tip: Always verify the relationship between edges and vertices when dealing with graph components; it can reveal constraints on the structure of the graph.