Broadcasts on Paths and Cycles

Abstract

A broadcast on a graph $G=(V,E)$ is a function $f:V⟶\{0,\dots ,\mathrm{diam}(G)\}$ such that $f(v)\le e\_G(v)$ for every vertex $v\in V$, where$\mathrm{diam}(G)$ 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 $\sum \_v\in Vf(v)$.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].