Covering Salesman Problem (CSP) polynomial reduction to the TSP
I am facing one problem that consists in polynomially reducing the Coverging Salesmen Problem (CSP) to the Traveling Salesman Problem (TSP). So, let me first define the CSP. The CSP, I am working on, can be defined as a complete undirected graph $G(V, E)$, a metric edge weight function $w_e \in \mathbb{R}^+$, and a node covering function $S: V \rightarrow \mathcal{P}(V)$ such that $i \in S(i)$, where $\mathcal{P}$ is the power set function. The CSP goal is to find a minimum cost tour that covers all the nodes in $G$. 835
For example, let's take a look at the CSP instance example in the Figure below, where $S(a) = S(b) = {a, b}$ and $S(c) = S(d) = {c, d}$.
A solution for this instance would be the tour $a, c$ as well as $b, d$, and if $w_{bc} = 2$, we could have the tour $b, c$, all of them with cost $4$.
So far I have planned a reduction scheme which I do not know if works but even though I would like to share with you, case it helps to show its correctness or find a working one. Let $\mathcal{S} = {S(i) : i \in V}$ be the set of clusters of the CSP instance, and for every $s \in \mathcal{S}$ let $C_s$ be an arbitrary cycle composed of the nodes in $s$, and $v_j^s \in s$ be the $j$-th node in $C_s$. We will convert $G$ into a digraph $D(N = {i, i^{'}, x_i : i \in V}, A = {(v_j^{s'}, v_j^s), (v_j^{s'}, x_{v_j^s}) : j \in \mathbb{N}^{}{\leqslant |s|}, s \in \mathcal{S}} \cup $ ${(x{v_j^s}, v_{j + 1}^{s'}) : j \in \mathbb{N}^{}{\leqslant |s| - 1}, s \in \mathcal{S}} \cup {(v_j^{s'}, v_k^t) : s \neq t \in \mathcal{S}, j \in \mathbb{N}^{}_{\leqslant |s|}, k \in \mathbb{N}^{}{\leqslant |t|}})$, and we will create an arc weight function $w^{'}a \in \mathbb{R}^{+}$ such that $w^{'}{(v_j^{s'}, v_j^s)} = w^{'}{(v_j^s, x{v_j^s})} = 0$ $\forall s \in \mathcal{S}, j \in \mathbb{N}^{}{\leqslant |s|}$, $w^{'}{(x_{v_j^s}, v_{j + 1}^s)} = 0$ $\forall s \in \mathcal{S}, j \in \mathbb{N}_{\leqslant |s| - 1}^{}$ , and $w^{'}{(v_j^{s'}, v_k^t)} = w{v_j^s, v_k^t}$ $\forall s \neq t \in \mathcal{S}, j \in \mathbb{N}^{}_{\leqslant |s|}, k \in \mathbb{N}^{}_{\leqslant |t|}$. The Figure below shows the resulting digraph for the aforementioned CSP instance.
My idea was to make all nodes in $C_s$ be visited by a single $i$-$i^{'}$-path $P \subset C_s$ $\forall s \in \mathcal{S}$. 835