$(t,r)$ Broadcast Domination Numbers and Densities of the Truncated Square Tiling Graph

TL;DR

This paper investigates broadcast domination numbers and densities in truncated square tiling graphs, providing bounds, exact values for specific cases, and constructions for infinite graphs to understand how signals can efficiently cover such tilings.

Contribution

It introduces new bounds and exact values for $(t,1)$ broadcast domination numbers in finite truncated square tiling graphs and constructs infinite broadcast schemes to estimate densities.

Findings

Exact $(2,1)$ broadcast domination numbers for small graphs.

Bounds for $(t,1)$ broadcast domination in finite graphs.

Constructed infinite broadcast schemes with density estimates.

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 $\sum_{v\in T, u\in N_t(v)}(t-d(u,v))\geq r$, where where $N_{t}(v)=\{u\in V:d(u,v)<t\}$ and $d(u,v)$ denotes the distance between $u$ and $v$. This can be interpreted as each vertex $v$ of $T$ sending $\max(t-\text{d}(u,v),0)$ signal to vertices within a distance of $t-1$ 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 $t\geq 1$) is the distance domination number of a graph. We study a family of graphs that arise as a finite subgraph of the truncated square titling, which utilizes regular squares and octagons to tile the Euclidean plane. For positive integers $m$ and $n$, we let $H_{m,n}$ be the graph consisting of $m$ rows of $n$ octagons (cycle graph on $8$ vertices). For all $t\geq 2$, we provide lower and upper bounds for the $(t,1)$ broadcast domination number for $H_{m,n}$ for all $m,n\geq 1$. We give exact $(2,1)$ broadcast domination numbers for $H_{m,n}$ when $(m,n)\in\{(1,1),(1,2),(1,3),(1,4),(2,2)\}$. We also consider the infinite truncated square tiling, denoted $H_{\infty,\infty}$, and we provide constructions of infinite $(t,r)$ broadcasts for $(t,r)\in\{(2,1),(2,2),(3,1),(3,2),(3,3),(4,1)\}$. Using these constructions we give upper bounds on the density of these broadcasts i.e., the proportion of vertices needed to $(t,r)$ broadcast dominate this infinite graph. We end with some directions for future study.