On Hardness and Approximation of Broadcasting in Structured Graphs

We study the Telephone Broadcasting problem in graphs with restricted structure. Given a designated source in an undirected graph, the goal is to disseminate a message to all vertices in the minimum number of rounds, where in each round every informed vertex may inform at most one neighbor. For general graphs, the problem is NP-hard. Recent work shows that the problem remains NP-hard even on restricted graph classes such as graphs of treewidth 2 [Tale 2025], cactus graphs of pathwidth 2 [Aminian et~al. 2025] and graphs at distance 1 to a path forest [Egami et~al. 2025]. In this work, we investigate the problem in several graph families. We first prove NP-hardness for cycle-star graphs, graphs formed by k cycles sharing a single vertex, as well as melon graphs, graphs formed by k paths with shared endpoints. Despite multiple efforts to understand the problem in these simple graph families, the computational complexity of the problem remained unsettled. Our hardness results answer open questions by Bhabak and Harutyunyan [2015] and Harutyunyan and Hovhannisyan [2023] concerning the problem's complexity in cycle-star and melon graphs, respectively. On the positive side, we present EPTASs for cycle-star and melon graphs, improving over the best existing approximation factors of 2 for both graph families. Moreover, we identify a structural frontier for tractability by showing that the problem is solvable in polynomial time on graphs of bounded cutwidth, a class that generalizes other families such as graphs of bounded bandwidth. This result subsumes existing tractability results for graph families such as necklace graphs. Finally, for split graphs, a fundamental class of highly structured graphs, we obtain a polynomial-time algorithm with approximation factor 1.76. This improves on the previously known factor 2 bound; the same approach also applies to the multi-source setting.