# Broadcasts on Paths and Cycles

## TL;DR

This paper determines various broadcast numbers for all paths and cycles in graphs, providing answers to previously open questions in the study of broadcast functions related to domination and independence.

## Contribution

It explicitly calculates broadcast numbers for paths and cycles, advancing understanding of broadcast functions in graph theory.

## Key findings

- Broadcast numbers for all paths and cycles are now explicitly known.
- The results resolve an open question from prior research.
- The study enhances the theoretical framework of broadcast functions in graphs.

## Abstract

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].