Efficient [formula omitted] broadcast dominating sets of the triangular lattice.

Blessing, Insko, Johnson and Mauretour gave a generalization of the domination number of a graph G called the (t , r) broadcast domination number which depends on the positive integer parameters t and r. In this setting, a v ∈ V is a broadcast vertex of transmission strength t if it transmits a signal of strength t − d (u , v) to every vertex u ∈ V with d (u , v) < t. Given a set of broadcast vertices S ⊆ V , the reception at vertex u is the sum of the transmissions from the broadcast vertices in S. The set S ⊆ V is called a (t , r) broadcast dominating set if every vertex u ∈ V has a reception strength r (u) ≥ 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 grid graph for all t ≥ r ≥ 1. Using these broadcasts, we then provide upper bounds for the (t , r) broadcast domination numbers for triangular matchstick graphs when (t , r) ∈ { (2 , 1) , (3 , 1) , (3 , 2) , (4 , 1) , (4 , 2) , (4 , 3) , (t , t) }. [ABSTRACT FROM AUTHOR]