the remaining receivers, respectively. Subsequently, we modify $\mathcal{G}$ to $\mathcal{G}^{\prime}$ and determine $\bold{\hat{X}}_{E^\prime}$ in $\mathcal{G}^{\prime}$ in order to ensure the validity of the corresponding network structure. Specifically, we determine $\bold{\hat{X}}_{E^\prime}$ by adding several messages to $\bold{\hat{X}}_{E}$ and determine $\{R_{e^\prime} | e^\prime\in E^\prime\}$ by adding a number of receivers to $\{R_{e} | e\in E\}$. Accordingly, some of the corresponding links of these added receivers are referred to as duplicated links, as explained later. Before choosing $\bold{\hat{X}}_{E^\prime}$, we classify problematic cases based on the outgoing edges of the receivers in $\mathcal{G}$, which should be modified to obtain $\mathcal{G}^\prime$.

There are six problematic cases based on message nodes and four problematic cases based on receiver nodes for the modification of the side information graph as in the following claim.

The problematic cases based on the outgoing edges of receivers in a side information graph $\mathcal{G}$ can be classified into the following ten cases, which should be modified to make the modified side information graph $\mathcal{G}^\prime$.

1.  $\hat{X}_s$ has one incoming edge from $\{R_t | t\in T\}$ for $s\in S$.

2.  $\hat{X}_s$ has more than one incoming edge from $\{R_t | t\in T\}$ for $s\in S$.

3.  $\hat{X}_s$ has one incoming edge from $\{R_{e^\prime} 