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.

翻译：本研究探讨了具有特定结构的图中的电话广播问题。给定一个无向图中指定的源节点，目标是在最少的轮数内将消息传播到所有顶点，其中每轮中每个已获知消息的顶点最多只能通知一个邻居。对于一般图，该问题是NP难的。近期研究表明，即使在受限图类上，该问题仍保持NP难，例如树宽为2的图[Tale 2025]、路宽为2的仙人掌图[Aminian等人 2025]以及距离路径森林为1的图[Egami等人 2025]。本文研究了若干图族中的该问题。首先证明了在环星图（由k个环共享单个顶点构成的图）以及甜瓜图（由k条共享端点的路径构成的图）上的NP难性。尽管已有多次尝试理解这些简单图族中的该问题，其计算复杂性一直悬而未决。我们的困难性结果分别回答了Bhabak与Harutyunyan[2015]以及Harutyunyan与Hovhannisyan[2023]关于该问题在环星图和甜瓜图上复杂性的公开问题。在积极方面，我们为环星图和甜瓜图提出了高效多项式时间近似方案，将现有最佳近似比从2进行了改进。此外，通过证明该问题在有界割宽图（此类图推广了其他图族如有界带宽图）上可在多项式时间内求解，我们确定了可处理性的结构边界。该结果涵盖了现有对项链图等图族的可处理性结论。最后，对于高度结构化的基本图类——分裂图，我们获得了近似比为1.76的多项式时间算法，改进了先前已知的2倍界；该方法同样适用于多源场景。