(t, r) broadcast domination numbers and densities of the truncated square tiling graph

Abstract For a pair of positive integer parameters ( t , r ), a subset T of vertices of a graph G is said to ( t , r ) broadcast dominate a graph G if, for any vertex u in G , we have $$\sum _{v\in T, u\in N_t(v)}(t-d(u,v))\ge r$$ ∑ v ∈ T , u ∈ N t ( v ) ( t - d ( u , v ) ) ≥ r , where $$N_{t}(v)=\{u\in V:d(u,v) N t ( v ) = { u ∈ V : d ( u , v ) < t } and d ( u , v ) denotes the distance between u and v . This can be interpreted as each vertex v of T sending $$\max (t-\text {d}(u,v),0)$$ max ( t - d ( u , v ) , 0 ) signal to vertices within a distance of $$t-1$$ t - 1 away from v . The signal is additive and we require that every vertex of the graph receives a minimum reception r from all vertices in T . For a finite graph, the smallest cardinality among all ( t , r ) broadcast dominating sets of a graph is called the ( t , r ) broadcast domination number. We remark that the (2, 1) broadcast domination number is the domination number and the ( t , 1) (for $$t\ge 1$$ t ≥ 1 ) is the distance domination number of a graph. We study a family of graphs that arise as a finite subgraph of the truncated square tiling, which utilizes regular squares and octagons to tile the Euclidean plane. For positive integers m and n , we let $$H_{m,n}$$ H m , n be the graph consisting of m rows of n octagons (cycle graph on 8 vertices). For all