Series-Parallel and Planar Graphs for Efficient Broadcasting

TL;DR

This paper constructs specific series-parallel and planar graphs with optimized broadcast times, advancing understanding of efficient information dissemination in these graph classes.

Contribution

It introduces new infinite families of series-parallel and planar graphs with optimized broadcast times, improving bounds and understanding of broadcast efficiency.

Findings

Constructed series-parallel graphs with broadcast time at most log2(n)+1

Developed planar broadcast graphs with improved vertex bounds

Enhanced lower bounds on planar broadcast graph size

Abstract

The broadcasting problem concerns the efficient dissemination of information in graphs. In classical broadcasting, a single originator vertex initially has a message to be transmitted to all vertices. Every vertex which has received the message informs at most one uninformed neighbor at each discrete time unit. In this paper, we introduce infinite families of series-parallel graphs with efficient broadcast times: graphs on $n$ vertices with broadcast time at most $\lceil\log_2 n \rceil + 1$ for any $n$, graphs on $n$ vertices with broadcast time $\lfloor \frac{3 \lceil \log_2 n \rceil}{2} \rfloor$ and maximum degree $\lceil \log_2 n \rceil - 1$ for any $n$, and broadcast graphs on up to $2^{k-1} + 2^{\lfloor \frac{k}{2} \rfloor }$ vertices with broadcast time $k$ for any $k$. We also introduce an infinite family of planar broadcast graphs on up to $2^{k-1} + 2^{\lfloor \frac{3k}{4} \rfloor - 1}$ vertices with broadcast time $k$ for any $k$, which improves the known lower bound on the maximum number of vertices in a planar broadcast graph.