Broadcasting under Structural Restrictions

TL;DR

This work studies the Telephone Broadcast problem on graphs with a designated source, formalizing $b(G,s)$ as the minimum number of rounds to inform all vertices. It proves NP-hardness on cactus graphs distance-1 to path forests and on graphs with constant tree-depth, and develops fixed-parameter algorithms parameterized by vertex integrity and distance to clique, along with a single-exponential DP for joint treewidth and rounds. It also delivers parameterized-approximation results for clique-cover and cluster-vertex-deletion parameters and establishes tight structural bounds linking $b(G,s)$ to pathwidth and tree-depth, notably improving the pathwidth-based approximation from $ ext{O}(4^{ ext{pw}})$ to $ ext{O}( ext{pw})$. These results map the tractability frontier of broadcasting under structural restrictions and provide practical, scalable approaches for sparse and certain dense graph families.

Abstract

In the Telephone Broadcast problem we are given a graph $G=(V,E)$ with a designated source vertex $s\in V$. Our goal is to transmit a message, which is initially known only to $s$, to all vertices of the graph by using a process where in each round an informed vertex may transmit the message to one of its uninformed neighbors. The optimization objective is to minimize the number of rounds. Following up on several recent works, we investigate the structurally parameterized complexity of Telephone Broadcast. In particular, we first strengthen existing NP-hardness results by showing that the problem remains NP-complete on graphs of bounded tree-depth and also on cactus graphs which are one vertex deletion away from being path forests. Motivated by this (severe) hardness, we study several other parameterizations of the problem and obtain FPT algorithms parameterized by vertex integrity (generalizing a recent FPT algorithm parameterized by vertex cover by Fomin, Fraigniaud, and Golovach [TCS 2024]) and by distance to clique, as well as FPT approximation algorithms parameterized by clique-cover and cluster vertex deletion. Furthermore, we obtain structural results that relate the length of the optimal broadcast protocol of a graph $G$ with its pathwidth and tree-depth. By presenting a substantial improvement over the best previously known bound for pathwidth (Aminian, Kamali, Seyed-Javadi, and Sumedha [arXiv 2025]) we exponentially improve the approximation ratio achievable in polynomial time on graphs of bounded pathwidth from $\mathcal{O}(4^\mathrm{pw})$ to $\mathcal{O}(\mathrm{pw})$.

Figures (2)

• Figure 1: Parameterized complexity of Telephone Broadcast with respect to structural graph parameters. The connection between two parameters represents that the upper parameter $p$ is bounded by some computable function $f(\cdot)$ of the lower parameter $q$, i.e., $p \le f (q)$. The parameters marked with $*$ are studied in this paper. The double, rounded, and dotted rectangles indicate paraNP-complete, fixed-parameter tractable, and parameterized approximable cases, respectively. The exact parameterized complexity of the parameterizations in the last case remains unsettled.
• Figure 5: The set $S = \{s,v\}$ is the unique set satisfying that $|S| + \max_{C \in \text{\tt cc}(G-S)} |V(C)| \le 7$ ($= \text{\rm vi}(G)$). In all optimal broadcast protocols, $v$ is the last vertex that receives the message from $s$ as $v$ needs only three rounds after it receives the message, while other neighbors needs four rounds.

Theorems & Definitions (9)

• Theorem 1
• Theorem 10
• Theorem 15
• Lemma 21
• Lemma 22
• Theorem 24
• Definition 25: arxiv/AminianKSS25
• Corollary 30
• Corollary 31