in which a side information VM exists at Node 3, while the side information $Y$ is known at both Node 1 and Node 2 and satisfies the Markov chain $X\textrm{---}Y\textrm{---}Z$. This characterization extends the result of [@Chia] discussed above to a model with a VM at Node 3. We remark that in [@Ahmadi_DSC], the rate-distortion-cost characterization for the model in Fig. \[fig:fig2\] was obtained, but under the assumption that the side information at Node 3 be available in a causal fashion in the sense of [@Weiss-Elgam];

-   *Cascade-broadcast source coding problem with VM at Node 2 and Node 3, lossless compression* (Fig. \[fig:fig3\]): In Sec. \[sub:RD\_BC\_lossless\], we study the cascade-broadcast model in Fig. \[fig:fig3\] in which a VM exists at both Node 2 and Node 3. In order to enable the action to be taken by both Node 2 and Node 3, we assume that the information about which action should be taken by Node 2 and Node 3 is sent by Node 1 on the broadcast link of rate $R_{b}$. Under the constraint of lossless reconstruction at Node 2 and Node 3, we obtain a characterization of the rate-cost performance. This conclusion generalizes the result in [@Weissman_multi] discussed above 