A degree sequence $d_1,d_2,d_3\ldots d_n$ of non negative integer is graphical if it is a degree sequence of a graph. We now introduce a powerful tool to determine whether a particular sequence is graphical due to Havel and Hakimi
Havel–Hakimi Theorem :
→ According to this theorem, Let $D$ be the sequence $d_1,d_2,d_3 \dots dn$ with $d_1 \geq d_2 \geq d_3 \geq \ldots d_n$ for $n \geq 2$ and $d_i \geq 0$.
→ If each $d_i = 0$ then $D$ is graphical
→ Then $D_0$ be the sequence obtained by:
→ Discarding $d_1$, and
→ Subtracting $1$ from each of the next $d_1$ entries of $D$.
→ That is Degree sequence $D_0$ would be : $d_2-1, d_3-1, \ldots , d_{d_1+1} -1, \ldots , d_n$
→ Then, $D$ is graphical if and only if $D_0$ is graphical.
Now, we apply this theorem to given sequences:
- $7,6,5,4,4,3,2,1 \rightarrow 5,4,3,3,2,1,0 \rightarrow 3,2,2,1,0,0 \rightarrow 1,1,0,0,0 \rightarrow 0,0,0,0$ so it is graphical.
- $6,6,6,6,3,3,2,2 \rightarrow 5,5,5,2,2,1,2$ ( arrange in descending order) $\rightarrow 5,5,5,2,2,2,1 \rightarrow 4,4,1,1,1,1 \rightarrow 3,0,0,0,1$ we cannot continue to get all $0's$, so it is not graphical.
- $7,6,6,4,4,3,2,2 \rightarrow 5,5,3,3,2,1,1 \rightarrow 4,2,2,1,1,0 \rightarrow 1,1,0,0,0 \rightarrow 0,0,0,0$ so it is graphical.
- $8,7,7,6,4,2,1,1$, here degree of a vertex is $8$ and total number of vertices are $8$, which is impossible, hence it is not graphical.
Hence, only option (I) and (III) are graphic sequence and answer is option-(D)
