On (t, r) Broadcast Domination of Certain Grid GraphsNatasha Crepeau

Harvey Mudd CollegeSean Hays

University of South FloridaAlexandro Vasquez

Manhattan College

AbstractIf G = 〈V (G), E(G)〉 is a connected graph, then a set D ⊂ V (G) is dominating if every vertex

of V − D has at least one neighbor in D. A generalization of this concept is (t, r) broadcast

domination. In this setting certain vertices are designated as towers of signal strength t, which

send out signal to neighboring vertices 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) as

f (v) =∑w∈Tmax(0, t− d(v, w)), where w is a tower and d(v, w) denotes the distance between

v and w. Blessing, Insko, Johnson, Mauretour (2014) 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

minimal cardinality of a (t, r) broadcast on G is called the (t, r) broadcast domination number

ofG. In this poster, we present our research on the (t, r) broadcast domination number for graphs

including paths, grid graphs, and different lattices.

IntroductionStructures such as cell phone towers radiate signal to their surrounding area. As the distance from

the tower increases, the signal becomes weaker. Eventually, areas far enough distance away will

be unable to receive any signal. As a result, the towers must be placed strategically to provide

signal to all areas while using as little towers as possible. The concept of (t, r) broadcasting

models this scenario.

1 2 3 2 1

Figure 1: Broadcast Tower of Strength 3 Sends Signal Out to Other Vertices

The domination number for (t, r), denoted γt,r(G), is the minimum cardinality of a dominating

set for G.

The concept of (t, r) broadcast domination is relatively new, as David Blessing et al. first intro-

duced it in 2014. There are a few notable results found thus far on grid graphs G3,n, and G4,n

for various (t, r). This poster improves on these results with explicit formulas for the domina-

tion number for (t, r) on G3,n and G4,n, as well as Gm,n and the path on n vertices. We then

show how the strategies used to find these explicit formulas can be extended to 3 dimension grid

graphs, slant lattices, and the king’s lattice.

t− 2

t

t

tr

rr

r

t− 2

t− 1t− 1t− 1

t− 2

Figure 2: 3× n Grid Graph Domination Pattern

Results

r r + 1 t t− 1 r r − 1 2 1

Figure 3: Beginning Construction of the Dominating Set of a Path

We found γt,r(Pn) by breaking the path into partitions of 2t− r. However, r− 1 vertices must be

receiving signal from 2 broadcast towers, which is how we arrive at Theorem 1.

Theorem 1If n ≥ 1 and t ≥ r ≥ 1, then

γt,r(Pn) =

⌈n + (r − 1)

2t− r

⌉.

Using what we learned about paths, we began looking at γt,r on a m× n grid graph. In order to

dominate the m× n grid we introduced the notion of a starting block, which is a grid efficiently

dominated by two towers. Lemma 1 describes the size of a starting block for the m× n grid.

Lemma 1. Let t ≥ r ≥ 1 and fix m ≥ 2. If n ≥ 2t− r − (m− 2) > 0, then Gm,n has a starting

block of size m× (2t− r − (m− 2)).

2t− r − (m− 2)

m

t

t

· · ·

..

.

· · ·......

Figure 4: m× 2t− r − (m− 2) Starting Block

Theorem 2If m ≥ 2, n ≥ 2t− r − (m− 2), and 2t− r > m− 1, then

γt,r(Gm,n) = 2 +

⌈n− (2t− r − (m− 2))

(2t− r − (m− 2))− 1

⌉.

Proof Sketch:

• Construct a block with two towers on opposite corners, then use the size of the path outlined

in red in Figure 4 to find the number of columns it has.

•Dominate the grid by breaking it up into the blocks we created.

If the grid has at least 1 starting block, we know there is at least 2 broadcasting towers in the

dominating set. So, we establish an initial starting block and then find how many more starting

blocks are contained in the grid. This is equivalent to placing a tower every 2t− r− (m− 2)− 1

column away from the previous broadcast tower, which is how we arrive at γt,r(Gm,n).

Extending Our ResultsWe were able to find the broadcast domination number for any (t, r) on a path and for many (t, r)

on the m× n grid graph. Similar strategies can be used to find γt,r for 3D grid graphs, the slant

lattice, and the king’s lattice. One application of the starting block strategy is to cover a 3D grid

graph in starting blocks, giving us an upper bound on γt,r.

m

n

k

Figure 5: A 3D Grid Graph with a Potential Starting Block

We can also construct starting blocks on the slant and king’s lattice, and use them to dominate

the rest of the lattice.

Figure 6: Creating a Starting Block for the King’s Lattice: (3,2) Example

References

[1] David Blessing, Erik Insko, Katie Johnson, and Katie Mauretour, On (t,r) Broadcast

Domination Numbers of Grids, arXiv e-prints (2014Jan), arXiv:1401.2499, available at

1401.2499.

AcknowledgmentsThis research was conducted during the 2019 Mathematical Sciences Research Institute Under-

graduate Program in Berkeley, CA under the direction of Dr. Pamela E. Harris (Williams Col-

lege). This program is supported by the Alfred P. Sloan Foundation, Mathematical Sciences

Research Institute, and the National Science Foundation. We would like to thank Prof. Rebecca

Garcia (Sam Houston State University), Dr. Marissa Loving (Georgia Institute of Technology),

Gordon Rojas Kirby (University of California at Santa Barbara), and Robert Joseph Rennie (Uni-

versity of Illinois Urbana-Champaign) for their mentorship.

Contact Information: Natasha Crepeau: ncrepeau@hmc.edu;

Sean Hays: seanhays@mail.usf.edu;

Alexandro Vasquez: avasquez03@manhattan.edu