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On $(t,r)$ broadcast domination of certain grid graphs

Natasha Crepeau, Pamela E. Harris, Sean Hays, Marissa Loving, Joseph Rennie, Gordon Rojas Kirby and Alexandro Vasquez

Vol. 16 (2023), No. 5, 883–903

Let G = (V (G),E(G)) be a connected graph with vertex set V (G) and edge set E(G). We say a subset D of V (G) dominates G if every vertex in V D is adjacent to a vertex in D. A generalization of this concept is (t,r) broadcast domination. We designate certain vertices to be towers of signal strength t, which send out signal to neighboring vertices with signal strength decaying linearly as the signal traverses the edges of the graph. We let 𝕋 be the set of all towers, and we define the signal received by a vertex v V (G) from all towers w 𝕋 to be f(v) = w𝕋 max (0,t d(v,w)). Blessing, Insko, Johnson and Mauretour defined a (t,r) broadcast dominating set, or a (t,r) broadcast, on G as a set 𝕋 V (G) such that f(v) r for all v V (G). The minimum cardinality of a (t,r) broadcast on G is called the (t,r) broadcast domination number of G. We present our research on the (t,r) broadcast domination number for certain graphs including paths, grid graphs, the slant lattice, and the king’s lattice.

graph domination, grid graphs
Mathematical Subject Classification
Primary: 05C12, 05C30, 05C69
Received: 23 August 2022
Accepted: 1 December 2022
Published: 9 December 2023

Communicated by Joel Foisy
Natasha Crepeau
Department of Mathematics
University of Washington
Seattle, WA
United States
Pamela E. Harris
Department of Mathematical Sciences
University of Wisconsin-Milwaukee
Milwaukee, WI
United States
Sean Hays
Department of Mathematics
The University of Alabama
Tuscaloosa, AL
United States
Marissa Loving
Department of Mathematics
University of Wisconsin-Madison
Madison, WI
United States
Joseph Rennie
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL
United States
Gordon Rojas Kirby
School of Mathematics and Statistical Sciences
Arizona State University
Tempe, AZ
United States
Alexandro Vasquez
Department of Mathematics
Manhattan College
Riverdale, NY 10471
United States