We consider communication of two degraded mes- sage sets over graphs where a common source sends two prioritized messages (a common and a private message) to several receivers. All receivers require the common message and a subset of the receivers require both the common and private messages. In this paper, we consider the case where all but two of the receivers require both messages. We provide an outer-bound on the rate region that depends on graph properties. We prove that this bound is achievable by using carefully selected linear operations at the network nodes. The achievability proof is built on our result in [14] and is illustrating potential connections of communication over deterministic channels and communication over graphs.