The domination number of a graph G=(V,E) is the minimum cardinality of
any subset SβV such that every vertex in V is in S or adjacent to
an element of S. Finding the domination numbers of m by n grids was an
open problem for nearly 30 years and was finally solved in 2011 by Goncalves,
Pinlou, Rao, and Thomass\'e. Many variants of domination number on graphs have
been defined and studied, but exact values have not yet been obtained for
grids. We will define a family of domination theories parameterized by pairs of
positive integers (t,r) where 1β€rβ€t which generalize domination
and distance domination theories for graphs. We call these domination numbers
the (t,r) broadcast domination numbers. We give the exact values of (t,r)
broadcast domination numbers for small grids, and we identify upper bounds for
the (t,r) broadcast domination numbers for large grids and conjecture that
these bounds are tight for sufficiently large grids.Comment: 28 pages, 43 figure