On $(t,r)$ broadcast domination of directed graphs

TL;DR

This paper extends the concept of broadcast domination to directed graphs, analyzing the range of domination numbers across all orientations and exploring specific graph classes like grids and stars.

Contribution

It introduces the directed $(t,r)$ broadcast domination framework and characterizes the possible domination numbers across all orientations of a graph.

Findings

For $r=1$ and $(t,r)=(2,2)$, all values in the interval are realizable as domination numbers.

The study provides bounds and existence results for domination numbers on various graph classes.

It opens avenues for future research in directed graph domination theory.

Abstract

A dominating set of a graph $G$ is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of $G$ is the order of a minimum dominating set of $G$. The $(t,r)$ broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength $t$ that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least $r$ from its set of broadcasting neighbors. In this paper, we extend the study of $(t,r)$ broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed $(t,r)$ broadcast domination numbers of all orientations of a graph $G$. In particular, we prove that in the cases $r=1$ and $(t,r) = (2,2)$, for every integer value in this interval, there exists an orientation $\vec{G}$ of $G$ which has directed $(t,r)$ broadcast domination number equal to that value. We also investigate directed $(t,r)$ broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.