Proving exact values for the $2$-limited broadcast domination number on grid graphs

We establish exact values for the 2-limited broadcast domination number of various grid graphs, in particular $C_m\square C_n$ for $3 \leq m \leq 6$ and all $n\geq m$, $P_m \square C_3$ for all $m \geq 3$, and $P_m \square C_n$ for $4\leq m \leq 5$ and all $n \geq m$. We also produce periodically optimal values for $P_m \square C_4$ and $P_m \square C_6$ for $m \geq 3$, $P_4 \square P_n$ for $n \geq 4$, and $P_5 \square P_n$ for $n \geq 5$. Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.


Introduction
Suppose there is a transmitter located at each vertex of a graph G.A k-limited broadcast f on G is a function f : V (G) → {0, 1, . . ., k}.The integer f (v) represents the strength of the broadcast from v, where f (v) = 0 means the transmitter at v is not broadcasting.A broadcast of positive strength f (v) from v is heard by all vertices u such that d(u, v) ≤ f (v), where d (u, v) is the distance between the u and v in G.A broadcast f is dominating if each vertex of G hears the broadcast from some vertex.The cost of a broadcast f is v∈V (G) f (v).The k-limited broadcast domination number γ b,k (G) of a graph G is the minimum cost of a k-limited dominating broadcast on G.
The k-limited broadcast domination number can be seen to be the optimum solution to ILP 1.1 shown below.Let G be a graph and fix 1 ≤ k ≤ rad(G), where rad(G) is the radius of G.For each vertex search of all possible small induced sub-broadcasts of given costs on a graph.Cases which provably cannot be part of an optimal broadcast are then eliminated.This approach can likely be extended to other graphs as well as general k-limited broadcast domination.Some intuition for our method is provided by example in Section 2. Section 3 describes and proves the correctness of the six schemes used to eliminate cases in the exhaustive search.Section 3 concludes with the statement of our main algorithm (Algorithm 2) to prove lower bounds and the proof of its correctness.Our results are summarized in Section 4. These include exact values for the 2-limited broadcast domination number of C m C n for 3 ≤ m ≤ 6 and all n ≥ m, P m C 3 for all m ≥ 3, and P m C n for 4 ≤ m ≤ 5 and all n ≥ m, and periodically optimal values for P m C 4 and P m C 6 for m ≥ 3, P 4 P n for n ≥ 4, and P 5 P n for n ≥ 5.These results improve upon the bounds in Slobodin's M.Sc.thesis Slobodin (2021).

Intuition and Definitions
This section includes a high-level overview of our method, an example specific to P 5 C n , and relevant definitions.
Definition 2.1.Let f be a 2-limited broadcast on the graph G and let X ⊆ V (G).Define the subbroadcast g induced by X by g(x) = f (x) if x ∈ X and 0 otherwise.
Throughout this paper, we consider a class of graphs G m,n equal to P m C n or C m C n for a fixed number of rows m.In this way, the vertex in the ith row and jth column can be denoted by (i, j).The goal is to prove that γ b,2 (G m,n ) is greater than or equal to a function B(m, n).We proceed by induction on n.After checking the appropriate base cases computationally, we assume the bound holds for all n < n 0 for some integer n 0 .Let f be a 2-limited dominating broadcast of G m,n0 .We choose values r and s such that, if the minimum cost (with respect to f ) of a sub-broadcast induced by r consecutive columns of G m,n0 is strictly greater than s, then B(m, n 0 ) ≤ γ b,2 (G m,n0 ).We then exhaustively enumerate (computationally) all possible sub-broadcast induced by r consecutive columns of G m,n0 of cost less than or equal to s.If it is possible to conclude that, for each possible sub-broadcast g, either g cannot be a sub-broadcast of an optimal 2-limited dominating broadcast on G m,n0 or g forces B(m, n 0 ) ≤ γ b,2 (G m,n0 ), then the desired bounds follows.See Example 2.2.
Proposition 2.3.(Slobodin et al., 2023, Theorem 3) Suppose we wish to obtain optimal values for γ b,2 (P 5 C n ) when n ≡ 0 (mod 2) by proving that n ≤ γ b,2 (P 5 C n ).We have that n ≤ γ b,2 (P 5 C n ) for 3 ≤ n ≤ 16 by computation.Suppose the bound holds for all n < n 0 for some n 0 > 16.Let f be a 2-limited dominating broadcast of P 5 C n0 .Let C be the subgraph of P 5 C n0 induced by the vertices appearing in a minimum cost (with respect to f ) set of eight consecutive columns (here r = 8).If the sub-broadcast induced by V (C) has cost at least 8 (here s = 7), then cost(f ) ≥ n 0 .It is therefore sufficient to consider, for each integer x ≤ 7, all possible sub-broadcasts of cost x induced by V (C).If it is possible to conclude that, for each such sub-broadcast g, either g cannot be a sub-broadcast of an optimal 2-limited dominating broadcast on P 5 C n0 or g forces n 0 ≤ γ b,2 (P 5 C n0 ), then the desired bounds follows.
We conclude this section with definitions used throughout the rest of the paper.
Definition 2.4.Given two 2-limited broadcasts f and g on a graph G, for each x ∈ V (G), define Definition 2.6.Let f be a broadcast on G.The broadcast range of f is the set of vertices which hear a broadcast under f .

Eliminating Possible Induced Sub-broadcasts of Fixed Cost
Fix m and suppose we wish to establish the function B(n) as a lower bound for the 2-limited broadcast domination number of G m,n .A positive fixed number r = r(m) ≥ 5 of columns is chosen.In the inductive step, we consider n 0 > r + 10 such that B(n) is a lower bound for γ b,2 (G m,n ) for all n < n 0 .Let C be the subgraph of G m,n0 induced by the vertices of r consecutive columns.We complete an exhaustive search of all possible sub-broadcasts g induced by V (C) and subject each such g to a series of tests in the hope of excluding g or concluding that g forces B(n 0 ) ≤ γ b,2 (G m,n0 ).Given C, four columns are added to both the left and right of C in order to ensure that the subgraph considered is large enough to include all vertices dominated by any vertex that could potentially dominate some vertex in C.
In summary, our algorithm takes as an input, H m,k = (P m or C m ) P k , where k = r + 8, with columns labelled c 1 , c 2 , . . ., c k , where the vertices of C are in columns c 5 , c 6 , . . ., c k−4 .Note that (P m or C m ) P k is understood to mean P m P k or C m P k dependent upon G m,n .Observe that k ≥ 13 and n 0 ≥ k + 3. The assumptions defined previously are used in Sections 3.1 through 3.6 which describe and prove the correctness of the six schemes we use to eliminate sub-broadcasts.These schemes appear in the same order as in Algorithm 2 in Section 3.7.

Domination Requirement
Since we are looking for a lower bound for the cost of an optimal 2-limited dominating broadcast f on G m,n0 , any induced sub-broadcast that forces vertices of G m,n0 to not be dominated, can be eliminated.
Observation 1.If the sub-broadcast g induced by V (C) does not dominate the vertices of columns c 7 , c 8 , . . ., c k−6 , then g cannot be a sub-broadcast of a dominating broadcast.
See Figure 1.The region containing V (C) is depicted by the thick black rectangle.The black circles with a black inner fill indicate vertices broadcasting at a non-zero strength.The thick red dotted lines indicate the broadcast ranges of the broadcasting vertices at their centers.In this example, there is one vertex broadcasting at strength 2 in columns c 4 and c k−3 and one vertex broadcasting at strength 1 in column c k−3 .Let DoesN otDominate(H m,k , g) return true if g does not dominate the vertices of columns c 7 , c 8 , . . ., c k−6 and false otherwise.

Forbidden Broadcasts
To improve the speed of our computations, we have identified four simple forbidden broadcast structures.Without loss of generality, a sub-broadcast g cannot contain any of the forbidden broadcasts shown in Figure 2 because the broadcast in a) cannot be found in an optimal 2-limited broadcast, and the broadcasts in b), c), and d) can be replaced by a broadcast of strength 2 from w while preserving the cost of g and extending the range of g.Let F orbiddenBroadcast(H m,k , g) return true if g exhibits a), b), c), or d) and false otherwise.

Optimality Requirement
As we are attempting to prove a lower bound for the cost of an optimal 2-limited dominating broadcast on G m,n0 , any possible sub-broadcast that is not optimal can be eliminated.
Observation 2. If the broadcast range R of a possible sub-broadcast g induced by V (C) can be dominated by a broadcast h on H m,k of cost strictly less than cost(g), then g cannot be a sub-broadcast of an optimal dominating broadcast.
Let HasBroadcast(H m,k , R, x) return true if R can be dominated with cost less than or equal to x on H m,k and false otherwise.

Proof by Induction
Recall that, after checking the appropriate base cases, we assume B(n) is a lower bound of γ b,2 (G m,n ) for all n < n 0 where n 0 ≥ k+3.Additionally, the values r = k−8 and s are chosen such that, if the minimum cost of a sub-broadcast induced by r consecutive columns of G m,n0 is strictly greater than s, then B(n 0 ) ≤ γ b,2 (G m,n0 ).As such, we may be able to conclude (via the inductive assumption) that possible subbroadcasts g of cost less than or equal to s induced by r consecutive columns of G m,n0 imply the bound we hope to prove.This can be done by deleting i ≤ k columns and "patching" the graph back together to obtain a 2-limited dominating broadcast on G m,n0−i , the cost of which we assumed to be greater than or equal to B(n 0 − i).In general, given some possible induced sub-broadcast g whose broadcast range R is contained within k consecutive columns, we delete a particular selection of i columns, for each i from 1 to k.To this end, define Algorithm 1 and Lemma 3.1 formalize our approach.See Example 3.2 for an illustration of this test on P 5 C n .
Algorithm 1: Routine to determine if assumed sub-broadcast implies bound.
) labelled according to their row number and column label, the cost x used to dominate R by some 2-limited broadcast g whose broadcast range lies entirely within H m,k , and m i (for each i = 1 to k) as defined in Section 3.4.Output: Value of the truth statement: "sub-broadcast implies bound." 2 Create a sorted list L of the columns that contain at least one vertex of R so that they are first ordered from maximum to minimum according to the number of vertices that are in R. Resolve ties by sorting so that c i comes before c j if i < j; 3 for i from 1 to length(L) do 4 Let S be the set of columns contained in the first i entries of list L; ).Here m i (for each i = 1 to k) is defined as in Section 3.4.
Proof: Assume the conditions of Lemma 3.1.Let f be an optimal 2-limited dominating broadcast of G m,n0 which contains g as an induced sub-broadcast.Let f ′ = f ⊖ g and let S, H m,k−|S| , and R ′ be defined as in Algorithm 1.Let G m,n0−|S| be constructed by removing the set of columns S from G m,n0 which resulted in InductiveArgument returning true on line 7 and adding edges in the natural way such that the resulting graph is isomorphic to e. all vertices v in the set of columns S) broadcasting with non-zero strength under f ′ , pick a vertex u in the same row as v and in an undeleted column nearest to v and let . There are two cases: Case I, v hears a broadcast from a vertex u under f ′ and u is not in the set of columns S.
Case II, v hears a broadcast from a vertex u under f ′ and u is in the set of columns S.
there are two subcases: Subcase II.a), u ′ is on the same column as v or u ′ is on the column between v and u on G m,n0 .Since u ′ is in the same row as u, d(u Subcase II.b), u is on a column between v and u ′ on G m,n0 .As u ′ is on a nearest column to u that is undeleted, the column of u ′ on G m,n0−|S| is at distance at most d Gm,n 0 (u, v) (the distance between u and v in G m,n0 ) from the column containing v. As u ′ is in the same row as u and By Observation 4, to dominate G m,n0−|S| , it suffices to find a 2-limited broadcast which dominates R ′ .As the function call on line 7 returns true, R ′ can be dominated by a 2-limited broadcast h with cost less than or equal to x − m |S| where x = cost(g).Note that, for this to be true, (2) Observe that (3) As m |S| ≥ B(n 0 ) − B(n 0 − |S|), when combined with Equations 2 and 3, we have that as desired.
Example 3.2.Suppose we wish to establish n ≤ γ b,2 (P 5 C n ); doing so will yield periodically optimal values for γ b,2 (P 5 C n ).By computation, we have that n ≤ γ b,2 (P 5 C n ) for 3 ≤ n ≤ 16.Suppose the bound holds for all n < n 0 where n 0 > 16.Let f be an optimal 2-limited dominating broadcast of P 5 C n0 and let C be the subgraph of P 5 C n0 induced by the vertices appearing in a minimum cost set of eight consecutive columns with respect to f .Suppose V (C) induces the sub-broadcast g shown in Figure 3.
The vertices of R ′ are indicated by the green circles in Figures 4 (Middle).The broadcast f ′′ dominates V (G m,n0−4 ) with the possible exception of the vertices of R ′ .However, R ′ can be dominated by the broadcast h of cost 3 as shown in Figure 4 Proving exact values for the 2-limited broadcast domination number on grid graphs 9

Necessary Broadcasts
Given some possible sub-broadcast g, the broadcast structure of g may imply the existence of a larger sub-broadcast g ′ .If this broadcast g ′ does not pass the optimality requirement (see Section 3.3), then g can be eliminated.If this broadcast g ′ allows for the induction argument (see Section 3.4), then g implies the bound we hope to prove.
Observation 5.As g is induced by V (C), for each vertex in column c 6 or c k−5 not dominated by g, any dominating broadcast of G m,n0 must have a vertex broadcasting at strength 2, in the same row and in column c 4 or c k−3 .
See Figure 5 (Left).Let g ′ be the broadcast constructed from g by adding these necessary broadcasts.Let R ′ be the broadcast range of g

Considering All Possible Sub-Cases
When considering all possible sub-broadcasts g and the necessary sub-broadcasts g ′ they imply (see Section 3.5), some sub-broadcasts may require that we consider all possible induced sub-broadcasts g ′′ which extend g ′ to dominate V (C).Let g ′ be the necessary broadcast implied by g as described in Section 3.5.Note g ′ dominates V (C) with the possible exception of vertices in columns c 5 and c k−4 .There are many possible ways a dominating 2-limited broadcast could dominate the vertices in columns c 5 and c k−4 undominated by g ′ .Let C ′ be the collection of broadcasts formed by extending g ′ to include vertices from )] broadcasting at strength 0, 1, or strength 2 until every vertex in columns c 5 and c k−4 which do not hear a broadcast under g ′ is dominated.See Figure 5 or InductiveArgument(H m,k , R ′′ , cost(g ′′ ), m 1 , m 2 , . . ., m k ) returns true for every g ′′ ∈ C ′ (where R ′′ is the broadcast range of g ′′ ) and false otherwise. of C are in columns c 5 , c 6 , . . ., c k−4 of H m,k and C ′ = H m,k−12 .The sub-broadcast g of f induced by V (C) must dominate C ′ (Observation 1).As s = γ b,2 (C ′ ), we have that cost(g) ≥ s.By our choice of If cost(g) > t then, by the definition of t, cost(f ) ≥ B(n) for all n ≥ n 0 .Thus, cost(g) ≤ t.As C defined on line 2 is the set of all possible sub-broadcasts of cost between s and t, inclusive, induced by the vertices of columns c 5 , c 6 , . . ., c k−4 , g ∈ C. As P rovedLowerBound returned true, one of the function calls on lines 4 through 10 returned true for g.From the results in Sections 3.1 through 3.6, this proves the claim.

Results
Our implementation includes a canonicity test for P m P n so as to only consider the set C of all possible sub-broadcasts induced by the vertices of some set of r consecutive columns with costs between s and t, inclusive, up to isomorphism.That is, for each pair of broadcasts g, g * ∈ C, there is no group action on P m P n which defines an automorphism between g and g * .This test was done by checking that each broadcast (when expressed as a sequence) was the maximum lexicographically when compared to all broadcasts isomorphic to it.When adapting the code to work on C m C n , we did not update the canonicity test to reduce the number of cases up to isomorphism on C m P n from P m P n .Fortunately, this redundancy was acceptable in terms of run time.The number of induced sub-broadcasts (i.e.|C|) have been verified by Pólya's Theorem (see (Brualdi, 2010, Theorem 14.3.3)).
Our implementation of P rovedLowerBound has allowed us to prove Theorems 4.1 through 4.11, and their respective corollaries.For each theorem, we include a table which summarizes the number of broadcasts rejected at each step of the algorithm per considered cost.Steps with zero cases are omitted.Additionally, as AllSubcases considers all possible induced sub-broadcasts of a given case, the total number of cases considered will be at least |C|.
Our implementation is written in C++ and available here Slobodin (2022).All ILP calls are run with a Gurobi solver Gurobi Optimization (2021).All computations in this section were run on Slobodin's 2021 16GB MacBook Pro with an Apple M1 Pro processor.
Running ProvedLowerBound for the above values took less than one second.
As r − 4 = 4 and γ b,2 (C 6 P 4 ) = 5, set s = 5.Let n 4 be the least residue of n modulo 4 and let c(n 4 ) be the constant in the upper bound dependent upon n 4 .Set t = 8.Observe that, for n = 19, thus t ≥ 8.If t > 8, then there exists an n > 18 such that which is a contradiction.As P rovedLowerBound (C 6 P 16 , 5, 8, m 1 , m 2 , . . ., m 16 ) is true, the result follows.
Running ProvedLowerBound for the above values took less than seven minutes.
The result follows from Theorem 4.11.

Future Work
This paper presents a method for computationally proving lower bounds for the 2-limited broadcast domination of the Cartesian product of two paths, a path and a cycle, and two cycles.Exact values for the 2-limited broadcast domination number of C m C n for 3 ≤ m ≤ 6 and all n ≥ m, P m C 3 for all m ≥ 3, and P m C n for 4 ≤ m ≤ 5 and all n ≥ m have been found, as have periodically optimal values for P m C 4 and P m C 6 for m ≥ 3, P 4 P n for n ≥ 4, and P 5 P n for n ≥ 5. Our method can likely be extended to other graphs and k-limited broadcast domination for k > 2. We note the follow rather natural questions.
Problem 5.1.Can this method be optimized further to prove bounds on larger graphs or graph other than the Cartesian product of two paths, a path and a cycle, and two cycles?
Problem 5.2.Can this method be altered to prove bounds for the k-limited broadcast domination number on the Cartesian product of two paths, a path and a cycle, and two cycles?
This computation took one year and considered over 223 trillion cases.For each proof in this paper, we used a backtracking algorithm to construct the set C of all possible sub-broadcasts.In our improved backtracking algorithm, we forbid the addition of any ForbiddenBroadcast (see Section 3.2).The number of cases produced by this improved backtrack cannot be verified by Pólya's Theorem (see (Brualdi, 2010, Theorem 14.3.3)).As such, these results will be reported elsewhere with an updated methodology and justification.

Fig. 1 :
Fig. 1: The graph H m,k with columns labelled c1, c2, . . ., c k , C indicated by the thick black rectangle, and possible broadcast vertices exterior to C which can only dominate vertices in columns c5, c6, c k−5 , and c k−4 .

Fig. 3 :
Fig. 3: Assumed sub-broadcast g induced by V (C) of cost 7.Let f ′ = f ⊖ g and let R be the range of g.The broadcast f ′ dominates V (P 5 C n0 ) with the possible exception of the vertices of R. Suppose we delete the four columns indicated in Figure4(Left) from the grid and add edges in the natural way such that the resulting graph G 5,n0−4 = P 5 C n0−4 .Let f ′′ be

Fig. 5 :
Fig. 5: The graph H m,k with columns labelled c1, c2, . . ., c k and C indicated by the thick black rectangle.(Left) Resulting necessary broadcasts of strength 2 in columns c4 and c k−3 forced by vertices undominated in columns c6 and c k−5 .(Right) Necessary broadcast g ′ as described in Section 3.5, the vertex undominated by g ′ indicated by the green circle, and one of the five possible sub-broadcasts g ′′ which extend g ′ to dominate the vertex undominated by g ′ in column c k−4 .

Proof:
The bound is easily verified by computation for 3 ≤ m ≤ 22. Theorem 5 of Slobodin et al. (2023) proves that γ b,2 (P m C 4 ) is less than or equal to the bound in the corollary statement for m ≥ 23.As any 2-limited dominating broadcast on P m C 4 is a 2-limited dominating broadcast on C m C 4 , γ b,2 (C 4 C m ) ≤ γ b,2 (P m C 4 ).The result follows from Theorem 4.3.Theorem 4.5.For n ≥ 5, γ b,2 (C 5 C n ) = n.
Tab. 1: Cases considered in the proof of Theorem 4.1.
4, then there exists an n > 16 such that Cases considered in the proof of Theorem 4.3.
Cases considered in the proof of Theorem 4.5.
Cases considered in the proof of Theorem 4.7.
Proving exact values for the 2-limited broadcast domination number on grid graphs 17 As r − 4 = 7 and γ b,2 (P 5 P 7 ) = 8, set s = 8.Let n 2 be the least residue of n modulo 2 and let c(n 2 ) be the constant in the upper bound dependent upon n 2 .Set t = 11.Observe that, for n = 23, = n + c(n 2 ) ≤ n + 1 ⇒ n < 11, which is a contradiction.As P rovedLowerBound (P 5 P 19 , 8, 11, m 1 , m 2 , . .., m 19 ) is true, the result follows.Running ProvedLowerBound for the above values took less than 30 minutes.Cases considered in the proof of Theorem 4.11.