Constructing bounded degree graphs with prescribed degree and neighbor degree sequences

Let $D=d_1,d_2,\dots ,d_n$ and $F=f_1,f_2,\dots ,f_n$ be two sequences of positive integers. We consider the following decision problems: is there a $i)$ multigraph, $ii)$ loopless multigraph, $iii)$ simple graph, $iv)$ connected simple graph, $v)$ tree, $vi)$ caterpillar $G=(V,E)$ such that for all $k$, $d(v_k)=d_k$ and ${\sum }_{w\in \mathcal{N}(v_k)}d(w)=f_k$ ($d(v)$ is the degree of $v$ and $\mathcal{N}(v)$ is the set of neighbors of $v$). Here we show that all these decision problems can be solved in polynomial time if ${\max }_k{d}_k$ is bounded. The problem is motivated by NMR spectroscopy of hydrocarbons.