becomes $\bold{\hat{Z}}G\neq\bold{0}$ for all $\bold{\hat{Z}}\in\mathcal{I}(q,\mathcal{G},\bm{\delta}_s)$, where $G$ is the corresponding generator matrix.

Let $\Phi$ be a set of subsets of $Z[n]$ defined by $$\Phi=\{B\subseteq Z[n]\big||\mathcal{X}_i\cap B|\geq2\delta_s^{(i)}+1 \textrm{ for all } i\in Z[m] \textrm{ s.t. } f(i)\cap B\neq \phi\}
\nonumber$$ for the side information graph $\mathcal{G}$ of a $(\bm{\delta}_s,\mathcal{G})$-ICSIE. Then, we have the following definition for a $\bm{\delta}_s$-cycle.

For a $(\bm{\delta}_s,\mathcal{G})$-ICSIE, a subgraph $\mathcal{G}^{\prime}$ of $\mathcal{G}$ is termed a $\bm{\delta}_s$-cycle if the set of message node indices of $\mathcal{G}^{\prime}$ is an element of $\Phi$ (i.e., $B$) and the set of user node indices of $\mathcal{G}^{\prime}$ consists of $i\in Z[m]$ such that $f(i)\in B$ and its edges consist of the corresponding edges in $\mathcal{G}$. The graph $\mathcal{G}$ is said to be $\bm{\delta}_s$-acyclic if there is no $\bm{\delta}_s$-cycle in $\mathcal{G}$. \[def:delta cycle\]

A $\bm{\delta}_s$-cycle is an important subgraph for a $(\bm{\delta}_s,\mathcal{G})$-ICSIE problem because the existence of the $\bm{\delta}_s$-cycle is a necessary and sufficient condition for the possibility to reduce its codelength. The following lemma shows the importance of the $\bm{\delta}_s$-cycle.

$\mathcal{G}$ is $\bm{\delta}_s$-acyclic if and only if $N_{\rm opt}^q(\bm{\delta}_s,\mathcal{G})=n$ for a $(\bm{\delta}_s,\mathcal{G})$-ICSIE, where $N_{\rm opt}^q(\bm{\delta}_s,\mathcal{G})$ is the optimal codelength.

The sufficiency part is similarly proved as in [@ICSIE] by showing the linear index 