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On (t,r) Broadcast Domination Numbers of Grids
The domination number of a graph is the minimum cardinality of
any subset such that every vertex in is in or adjacent to
an element of . Finding the domination numbers of by 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 where which generalize domination
and distance domination theories for graphs. We call these domination numbers
the broadcast domination numbers. We give the exact values of
broadcast domination numbers for small grids, and we identify upper bounds for
the broadcast domination numbers for large grids and conjecture that
these bounds are tight for sufficiently large grids.Comment: 28 pages, 43 figure
Asymptotically Optimal Bounds for (t,2) Broadcast Domination on Finite Grids
Let be a graph and be positive integers. The \emph{signal}
that a tower vertex of signal strength supplies to a vertex is
defined as where denotes the
distance between the vertices and . In 2015 Blessing, Insko, Johnson,
and Mauretour defined a \emph{ broadcast dominating set}, or simply a
\emph{ broadcast}, on as a set such that the
sum of all signals received at each vertex from the set of towers
is at least . The broadcast domination number of a
finite graph , denoted , is the minimum cardinality over
all broadcasts for .
Recent research has focused on bounding the broadcast domination
number for the grid graph . In 2014, Grez and Farina
bounded the -distance domination number for grid graphs, equivalent to
bounding . In 2015, Blessing et al. established bounds
on , , and
. In this paper, we take the next step and provide a
tight upper bound on for all . We also prove the
conjecture of Blessing et al. that their bound on is
tight for large values of and .Comment: 8 pages, 4 figure
Asymptotically Optimal Bounds for (t,2) Broadcast Domination on Finite Grids
Let G = (V,E) be a graph and t,r be positive integers. The signal that a tower vertex T of signal strength t supplies to a vertex v is defined as sig(T, v) = max(t − dist(T,v),0), where dist(T,v) denotes the distance between the vertices v and T. In 2015 Blessing, Insko, Johnson, and Mauretour defined a (t, r) broadcast dominating set, or simply a (t, r) broadcast, on G as a set T ⊆ V such that the sum of all signal received at each vertex v ∈ V from the set of towers T is at least r. The (t, r) broadcast domination number of a finite graph G, denoted γt,r(G), is the minimum cardinality over all (t,r) broadcasts for G.
Recent research has focused on bounding the (t, r) broadcast domination number for the m×n grid graph Gm,n. In 2014, Grez and Farina bounded the k-distance domination number for grid graphs, equivalent to bounding γt,1(Gm,n). In 2015, Blessing et al. established bounds on γ2,2(Gm,n), γ3,2(Gm,n), and γ3,3(Gm,n). In this paper, we take the next step and provide a tight upper bound on γt,2(Gm,n) for all t \u3e 2. We also prove the conjecture of Blessing et al. that their bound on γ3,2(Gm,n) is tight for large values of m and n
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