Broadcasts on Paths and Cycles

A broadcast on a graph $G=(V,E)$ is a function $f: V\longrightarrow \{0,\ldots,\operatorname{diam}(G)\}$ such that $f(v)\leq e_G(v)$ for every vertex $v\in V$, where$\operatorname{diam}(G)$ denotes the diameter of $G$ and $e_G(v)$ the eccentricity of $v$ in $G$. The cost of such a broadcast is then the value $\sum_{v\in V}f(v)$.Various types of broadcast functions on graphs have been considered in the literature, in relation with domination, irredundence, independenceor packing, leading to the introduction of several broadcast numbers on graphs.In this paper, we determine these broadcast numbers for all paths and cycles, thus answering a questionraised in [D.~Ahmadi, G.H.~Fricke, C.~Schroeder, S.T.~Hedetniemi and R.C.~Laskar, Broadcast irredundance in graphs. {\it Congr. Numer.} 224 (2015), 17--31]

A LINEAR-TIME ALGORITHM FOR BROADCAST DOMINATION IN A TREE

The broadcast domination problem is a variant of the classical minimum dominating set problem in which a transmitter of power p at vertex v is capable of dominating all vertices within distance p from v. Our goal is to assign a broadcast power f(v) to every vertex v in a graph such that the sum for all v over V of f(v) is minimized, and such that every vertex u with f(u) = 0 is within distance f(v) of some vertex v with f(v) \u3e 0. The problem is solvable in polynomial time on a general graph, and Blair et al. gave an O(n^2) algorithm for trees. We provide an O(n) algorithm for trees. Our algorithm is notable because it makes decisions for each vertex v based on \u27non-local\u27 information from vertices far away from v, whereas almost all other linear-time algorithms for trees only make use of local information

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum