A note on the lower bound of centralized radio broadcasting for planar reachability graphs

We consider deterministic radio broadcasting in radio networks whose nodes have full topological information about the network. We focus on radio networks having an underlying reachability graph planar. We show that there are reachability graphs such that it is impossible to complete broadcasting in less than 2.ecc(s,G) rounds, where ecc(s,G) is the eccentricity of a distinguished source ss of a graph GG. They provide a negative answer to the open problem stated by Manne, Wanq, and Xin in [F. Manne, S. Wanq, Q. Xin, Faster radio broadcasting in planar graphs, in: Proceedings of the 4th Annual Conference on Wireless on Demand Network Systems and Services, WONS’2007, IEEE Press, 2007, pp. 9–13] and show that it is impossible to complete the broadcasting task in D+O(1)D+O(1) rounds for each planar reachability graph. Particularly, we propose 3/2.D3/2.D lower bound with respect to the parameter DD — the diameter of a reachability graph. It is a nontrivial lower bound of time of centralized radio broadcasting in the case that an underlying reachability graph is planar. Moreover, we describe a generalized construction which can potentially improve the presented lower bound.