(t,r) Broadcast Domination of the Truncated Square Tiling Graph

Jillian Cervantes
Graduate Student
University of Wisconsin – Milwaukee

This talk will introduce graph domination theory and a generalization called (t,r) broadcast domination. 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 Hm,n be the graph consisting of m rows of n octagons (cycle graph on 8 vertices). For all t ≥ 2, we provide lower and upper bounds for the (t, 1) broadcast domination number for Hm,n for all m, n ≥ 1. We give exact (2, 1) broadcast domination numbers for Hm,n when (m, n) ∈ {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2)}. We also consider the infinite truncated square tiling, and we provide constructions of infinite (t, r) broadcasts for (t, r) ∈ {(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