consists of $\{R_e | e\in E\}$ and $\{\hat{X}_e | e\in E\}$.

2.  Each element of $\{\hat{X}_e | e\in E\}$ should be wanted by one receiver.

3.  If $R_i$ has $\hat{X}_s$ as side information for $s\in S$, $R_i$ cannot have $\hat{X}_e$ as side information for $e\in E$.

If a network structure is valid, the network structure is directed acyclic and the source nodes are not intermediate. Thus, to be directed acyclic, 1) should be satisfied. 2) is due to our setting of the relation and 3) should be satisfied because the source nodes are not intermediate nodes.

\[prop:condition\]

From Proposition \[prop:condition\], we note that the corresponding models do not cover all index coding instances but cover some index coding instances satisfying conditions in the above proposition necessarily. However, it is important to note that all network structures can be covered by this model.

In this model, one difference from that in [@EQU] is the existence of the receiver $\hat{t}_{\rm all}$ which can be described as $\mathcal{X}_{\hat{t}_{\rm all}}=S$, $f(\hat{t}_{\rm all})=E$, and $\delta_s^{(\hat{t}_{\rm all})}=0$. In fact, the existence of $\hat{t}_{\rm all}$ in [@EQU] originates from a directed acyclic network structure. Thus, an identical result can be obtained even if we remove $\hat{t}_{\rm all}$ 