Modelling simultaneous broadcasting by level-disjoint partitions

Simultaneous broadcasting of multiple messages from the same source vertex in synchronous networks is considered under restrictions that each vertex receives at most one message in a unit time step, every received message can be sent out only in the next time step, no message is sent to already informed vertex. The number of outgoing messages in unrestricted, messages have unit length, and we assume full-duplex mode. In [9] we developed a concept of level-disjoint partitions to study simultaneous broadcasting under this model. In this work we consider the optimal number of level-disjoint partitions. We also provide a necessary condition in terms of eccentricity and girth on existence of $k$ $v$-rooted level-disjoint partitions of optimal height. In particular, we provide a structural characterization of graphs admitting two level-disjoint partitions with the same root.