Let's break down the problem step by step.

Part (a): Draw a graph $G$ with degree sequence $4, 3, 3, 2, 2$

Steps to construct the graph:

1. We have 5 vertices labeled, say $v_1, v_2, v_3, v_4, v_5$.
2. The degree sequence means that vertex $v_1$ has degree 4, $v_2$ and $v_3$ have degree 3, and $v_4$ and $v_5$ have degree 2.
3. Since vertex $v_1$ has degree 4, it must be connected to all other vertices.
4. Now connect $v_2$ and $v_3$ to ensure they have degree 3, but ensure that $v_4$ and $v_5$ maintain degree 2 by carefully adding edges.

Here’s one possible graph:

• $v_1$ connects to $v_2, v_3, v_4, v_5$ (giving $v_1$ degree 4).
• $v_2$ connects to $v_3$ and $v_4$ (now $v_2$ has degree 3).
• $v_3$ connects to $v_5$ (now $v_3$ has degree 3, and both $v_4$ and $v_5$ have degree 2).

Part (b): Degree sequence of the complement of $G$

In the complement of a graph, two vertices are adjacent if and only if they were not adjacent in the original graph.

1. In the original graph, $v_1$ had degree 4, meaning it was connected to 4 vertices. In the complement graph, $v_1$ will be connected to 0 vertices (since it was connected to everyone).
2. Vertices $v_2$ and $v_3$ had degree 3 in the original graph, so in the complement, each will be connected to 1 vertex.
3. Vertices $v_4$ and $v_5$ had degree 2 in the original graph, so in the complement, each will be connected to 3 vertices.

Thus, the degree sequence of the complement graph is: $0, 1, 1, 3, 3$

Part (c): Prove or disprove that the given statement is a tautology using a truth table

A tautology is a statement that is always true regardless of the truth values of the components.

Let the given statement be $P$. Without the explicit statement provided, let's assume it's a logical formula involving variables $p, q$, and so on. I will guide you through the steps to test it using a truth table.

1. Identify the logical operators (e.g., AND, OR, NOT, implications).
2. Create a truth table for all possible truth values of the involved variables.
3. Evaluate the statement for each combination of truth values.
4. Check if the final column is all TRUE—if it is, the statement is a tautology; otherwise, it is not.

If you provide the exact statement, I can construct the truth table and proceed with the proof.

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Would you like further details or assistance on any of these steps?

Relative Questions:

1. Can you describe how the degree sequence helps construct a graph?
2. What are the properties of the complement of a graph?
3. How would the degree sequence of a graph change if a vertex were removed?
4. Why are truth tables important in proving logical statements?
5. How does the degree of vertices affect the structure of a graph?

Tip:

When constructing graphs from a degree sequence, ensure that each step of adding edges keeps the overall degree sequence intact for all vertices involved.