Graphs with prescribed degree sequences. Given a list of $n$ positive integers $d_{1}, d_{2}, \ldots, d_{n},$ we want
to efficiently determine whether there exists an undirected graph $G=(V, E)$ whose nodes have degrees precisely $d_{1}, d_{2}, \ldots, d_{n} .$ That is, if $V=\left\{v_{1}, \ldots, v_{n}\right\},$ then the degree of $v_{i}$ should be exactly $d_{i} .$ We call $\left(d_{1}, \ldots, d_{n}\right)$ the degree sequence of $G .$ This graph $G$ should not contain self-loops (edges with both endpoints equal to the same node) or multiple edges between the same pair of nodes.
(a) Give an example of $d_{1}, d_{2}, d_{3}, d_{4}$ where all the $d_{i} \leq 3$ and $d_{1}+d_{2}+d_{3}+d_{4}$ is even, but for which no graph with degree sequence $\left(d_{1}, d_{2}, d_{3}, d_{4}\right)$ exists.
(b) Suppose that $d_{1} \geq d_{2} \geq \cdots \geq d_{n}$ and that there exists a graph $G=(V, E)$ with degree sequence $\left(d_{1}, \ldots, d_{n}\right) .$ We want to show that there must exist a graph that has this degree sequence and where in addition the neighbors of $v_{1}$ are $v_{2}, v_{3}, \ldots, v_{d_{1}+1} .$ The idea is to gradually transform $G$ into a graph with the desired additional property.
i. Suppose the neighbors of $v_{1}$ in $G$ are not $v_{2}, v_{3}, \ldots, v_{d_{1}+1} .$ Show that there exists $i < $ $j \leq n$ and $u \in V$ such that $\left\{v_{1}, v_{i}\right\},\left\{u, v_{j}\right\} \notin E$ and $\left\{v_{1}, v_{j}\right\},\left\{u, v_{i}\right\} \in E$.
ii. Specify the changes you would make to $G$ to obtain a new graph $G^{\prime}=\left(V, E^{\prime}\right)$ with the same degree sequence as $G$ and where $\left(v_{1}, v_{i}\right) \in E^{\prime}$.
iii. Now show that there must be a graph with the given degree sequence but in which $v_{1}$ has neighbors $v_{2}, v_{3}, \ldots, v_{d_{1}+1}$.
(c) Using the result from part (b), describe an algorithm that on input $d_{1}, \ldots, d_{n}$ (not necessarily sorted) decides whether there exists a graph with this degree sequence. Your algorithm should run in time polynomial in $n$ and in $m=\sum_{i=1}^{n} d_{i}$.