Broadcasts on Paths and Cycles

Sabrina Bouchouika (1) , Isma Bouchemakh (1) , Eric Sopena (2)

1 L'IFORCE - L'IFORCE
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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, independence or 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 question raised 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].

Mots clés: Dominating broadcast; Broadcast; Irredundant broadcast; Cycle.; Independent broadcast; Packing broadcast; Path

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Sabrina Bouchouika, Isma Bouchemakh, Eric Sopena. Broadcasts on Paths and Cycles. Discrete Applied Mathematics, 2020, 283, pp.375-395. ⟨10.1016/j.dam.2020.01.030⟩. ⟨hal-02151205v2⟩