Modelling simultaneous broadcasting by level-disjoint partitions

TL;DR

This paper investigates the optimal number of level-disjoint partitions for simultaneous broadcasting in synchronous networks, providing structural characterizations and necessary conditions related to graph properties.

Contribution

It introduces a detailed analysis of the maximum number of level-disjoint partitions and offers structural characterizations and necessary conditions for their existence.

Findings

Derived necessary conditions based on eccentricity and girth.

Provided structural characterizations for graphs with two level-disjoint partitions.

Analyzed the optimal height of such partitions.

Abstract

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.