Broadcast Domination of Triangular Matchstick Graphs and the Triangular Lattice

TL;DR

This paper investigates the $(t,r)$ broadcast domination problem on infinite triangular grids, constructing efficient broadcast sets and deriving upper bounds for specific finite matchstick graphs, advancing understanding of signal coverage in triangular lattice structures.

Contribution

It introduces the construction of efficient $(t,r)$ broadcast dominating sets on infinite triangular lattices and provides new upper bounds for finite triangular matchstick graphs for various $(t,r)$ parameters.

Findings

Constructed efficient $(t,r)$ broadcast sets for all $t \\geq r \\geq 1$ on the infinite triangular lattice.

Derived upper bounds for the $(t,r)$ broadcast domination numbers of specific finite triangular matchstick graphs.

Extended the theory of broadcast domination to triangular lattice structures with practical bounds.

Abstract

Blessing, Insko, Johnson and Mauretour gave a generalization of the domination number of a graph $G=(V,E)$ called the $(t,r)$ broadcast domination number which depends on the positive integer parameters $t$ and $r$. In this setting, a vertex $v \in V$ is a broadcast vertex of transmission strength $t$ if it transmits a signal of strength $t-d(u,v)$ to every vertex $u \in V$, where $d(u,v)$ denotes the distance between vertices $u$ and $v$ and $d(u,v) <t$. Given a set of broadcast vertices $S\subseteq V$, the reception at vertex $u$ is the sum of the transmissions from the broadcast vertices in $S$. The set $S \subseteq V$ is called a $(t,r)$ broadcast dominating set if every vertex $u \in V$ has a reception strength $r(u) \geq r$ and for a finite graph $G$ the cardinality of a smallest broadcast dominating set is called the $(t,r)$ broadcast domination number of $G$. In this paper, we consider the infinite triangular grid graph and define efficient $(t,r)$ broadcast dominating sets as those broadcasts that minimize signal waste. Our main result constructs efficient $(t,r)$ broadcasts on the infinite triangular lattice for all $t\geq r\geq 1$. Using these broadcasts, we then provide upper bounds for the $(t,r)$ broadcast domination numbers for triangular matchstick graphs when $(t,r)\in\{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3),(t,t)\}$.