Series-Parallel and Planar Graphs for Efficient Broadcasting

TL;DR

This work addresses the problem of efficient information dissemination (broadcasting) in graphs by constructing infinite families of series-parallel and planar graphs with provably small broadcast times. The authors leverage binomial-tree based structures (BT_k, MB_k, B_k, EB_k, AB_k) and planar embeddings to design explicit broadcast schemes that approach or attain the information-theoretic lower bound. Major results include: (i) for all n, SP graphs with broadcast time at most $\lceil\log_2 n\rceil+1$; (ii) $MB_k$ on $n=2^k$ with $b=\left\lfloor\frac{3k}{2}\right\rfloor$ and degree bound $\lceil\log n\rceil-1$; (iii) $EB_k$ on $2^{k-1}+2^{\lfloor k/2\rfloor}$ vertices achieving $b=k$; (iv) planar graphs on up to $2^{k-1}+2^{\lfloor 3k/4\rfloor-1}$ vertices with $b=k$. The paper also proves a negative result: no SP broadcast graph exists on $2^k$ vertices for $k\ge3$, and introduces accelerated-planar variants (AB_k) to push planar-broadcast constructions further, advancing understanding of how topology and degree influence broadcast efficiency. These constructions provide practical templates for fault-tolerant, low-degree networks with near-optimal broadcasting performance.

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.

Figures (7)

• Figure 2: A binomial tree $BT_4$ is a root vertex with children $BT_3$, $BT_2$, $BT_1$, $BT_0$. The blue edges show that a $BT_4$ is also a $BT_2$ where each vertex is the root of an appended $BT_2$ (left). A $BT_4$ alternatively consists of two instances of $BT_3$ connected at their roots (right).
• Figure 3: A Mirrored-Binomial SP graph $MB_k$ can be seen as: an upper and lower $BT_{k-1}$ joined at the leaves, or alternatively, two $MB_{k-1}$ side by side, joined at their terminals. Since $dist(s, t') = k$, the path from $u$ to $s$ (in red) is of length $\lceil \frac{k}{2}\rceil +j$, and the path from $u$ to $t'$ (in blue) is of length $\lfloor \frac{k}{2}\rfloor -j$.
• Figure 4: An SP graph on $n = 2^k$ vertices with broadcast time $\lceil\log n\rceil + 1$.
• Figure 5: A Binomial SP graph $B_3$ on $2^3 + 1$ vertices generated from a $B_2$ using a sequence of SP operations.
• Figure 6: Beginning of the broadcast scheme for a vertex $v$ (in red) in an Extended-Binomial SP graph $EB_k$. At time 2, the originator begins informing the $BT_{\lfloor\frac{k}{2}\rfloor}$ subtree that it is contained in (the red triangle). Meanwhile at each time $i$, vertex $t$ informs $r_{k-i}$, the root of the $BT_{k-i}$ subtree that does not contain $v$.

Theorems & Definitions (23)

• Lemma 2.1
• proof
• Definition 3.1
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.5
• proof
• Theorem 3.6