To solve the problem of constructing or explaining why a graph cannot exist based on the given degree sequences, let’s analyze each case:

(a) Graph with four vertices of degrees $1, 2, 3, 3$:

1. Check the Handshaking Lemma: The sum of the degrees of the vertices must be even because every edge is counted twice in the degree sequence. Here: $1 + 2 + 3 + 3 = 9$ Since $9$ is odd, this violates the Handshaking Lemma. Thus, no such graph exists.

(b) Graph with four vertices of degrees $1, 1, 1, 4$:

1. Check the Handshaking Lemma: The sum of the degrees is: $1 + 1 + 1 + 4 = 7$ Since $7$ is odd, this also violates the Handshaking Lemma. Therefore, no such graph exists.

(c) Graph with four vertices of degrees $1, 2, 3, 4$:

1. Check the Handshaking Lemma: The sum of the degrees is: $1 + 2 + 3 + 4 = 10$ This is even, so it satisfies the Handshaking Lemma.

2. Construct the Graph:

   • Start with a vertex of degree $4$. Label it $A$, and connect it to all three other vertices $B, C, D$.
   • Assign degrees to $B, C, D$ such that:
     • $B$ has degree $3$ (already connected to $A$, connect $B$ to $C$ and $D$).
     • $C$ has degree $2$ (already connected to $A$ and $B$).
     • $D$ has degree $1$ (already connected to $A$).

   This satisfies the degree sequence $1, 2, 3, 4$, and such a graph exists.

Summary:

• (a): No such graph exists (violates the Handshaking Lemma).
• (b): No such graph exists (violates the Handshaking Lemma).
• (c): Such a graph exists, and it can be constructed as described.

Would you like me to draw the graph for part (c) or explain anything in more detail? Here are some related questions for exploration:

1. What is the Handshaking Lemma, and why is it crucial in graph theory?
2. Can you verify if the degree sequence $3, 3, 3, 3$ is possible for a graph?
3. How do you determine if a given degree sequence corresponds to a simple graph?
4. What is the importance of even degree sums in graph construction?
5. How can the Havel-Hakimi algorithm help determine graph realizability?

Tip: When analyzing graphs, always start with the Handshaking Lemma to save time on impossible constructions!