Domination broadcast: A case study on a combination of cycle graph and sunlet graph

TL;DR

This paper investigates the minimum cost of broadcast signal distribution in specific graph structures, providing proofs for cycle and sunlet graphs, and extending results to a generalized sunlet graph with degree n.

Contribution

It offers new proofs for the b3_bb1-dominating broadcast number in cycle and sunlet graphs, and extends the analysis to a generalized sunlet graph with degree n.

Findings

Established the b3_bb1-dominating broadcast number for cycle graphs.

Determined the b3_bb1-dominating broadcast number for sunlet graphs.

Extended the results to a generalized sunlet graph with degree n.

Abstract

Domination in graphs has long been studied and is applied to signal distribution problem. For example, telecommunication companies want to spread the signal from broadcast stations by transmitting varying signal strength to all receiving stations. This problem can be interpreted in a term of graph theory. Assume that broadcast companies need to spread the signal in graphs by using broadcast stations with varying signal strength so that the signal could be sent to all the vertices. However, broadcast stations with stronger signal are generally more expensive. The distribution of signal was configured to cover all stations with minimum total cost of signal called the $\gamma_b-$dominating broadcast number. This paper shows another proof of the $\gamma_b-$dominating broadcast number of cycle graphs and sunlet graphs as a foundation for the further result. I also consider the $\gamma_b-$dominating broadcast number of a generalized version of sunlet graphs whose vertices on the cycle are equally extended by the path, called the sunlet graph with degree $n$. To obtain the optimal cost of the signal distribution for this extended version, we show that it is sufficient to use only one broadcast station at a vertex on the cycle with the signal cost equal to the radius of the sunlet graph with degree $n$.