Download this article
 Download this article For screen
For printing
Recent Issues

Volume 16
Issue 5, 727–903
Issue 4, 547–726
Issue 3, 365–546
Issue 2, 183–364
Issue 1, 1–182

Volume 15, 5 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Author Index
Coming Soon
 
Other MSP Journals
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
Abstract

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.

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

Communicated by Joel Foisy
Authors
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