show that $\underline{C}^{iii}\leq\underline{C}^{ii}$ and thus $\underline{C}^{iii}$ will be excluded. The upper bound $\overline{C}_g$ can also be excluded. Then, the sum-capacity is bounded as follows $$\begin{aligned}
\underline{C}\triangleq\max\{\underline{C}^i,\underline{C}^{ii}\}\leq C_g\leq\min\{\overline{C}_{cs},\overline{C}_s\}\triangleq\overline{C}.\end{aligned}$$ Now, bounding the difference between $\overline{C}$ and $\underline{C}$ is a simple task, and we can show that $$\begin{aligned}
\overline{C}-\underline{C}\leq1,\end{aligned}$$ for any value of $P$. Figure \[Bounds\_R\] shows the upper and lower bounds for a symmetric Y-channel with $P=P_r$, where it can be seen that the gap is always less than 1 bit.

\[t\][$P$(dB)]{} \[b\][Sum Rate]{} \[bl\][[$\overline{C}$]{}]{} \[Bl\][[$\underline{C}$]{}]{} ![A plot of the upper and lower bounds for the symmetric Y-channel showing the 4 cases difference cases (1-4)[]{data-label="Bounds_R"}](sum_rate2.eps "fig:"){width="\columnwidth"}

Upper Bounds for the Restricted Y-channel {#Section:RYC}
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In this section, we impose an additional constraint on the Y-channel. That is, we consider the Y-channel with a restricted encoder (\[SourceEncoder\_R\]). Recall that the difference between the restricted Y-channel and the general one is that the transmit signals are independent in the former while they can be dependent in the later.

The independence of the transmit signals can lead to tighter upper bound. Namely, the upper bound in Theorem \[SRUBG\] can be tightened leading to a smaller gap to the lower bound. We start with the following 