A dominating set of a graph
G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of
G is the order of a minimum dominating set of
G. The (
t,
r) broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength
t that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least
r from its set of broadcasting neighbors. In this paper, we extend the study of (
t,
r) broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed (
t,
r) broadcast domination numbers of all orientations of a graph
G. In particular, we prove that in the cases
r = 1 and
(
t
,
r
) = (2, 2), for every integer value in this interval, there exists an orientation
Throughout this work we let G denote a finite simply connected graph, and let V ( G ) and E ( G ) denote its vertex and edge set, respectively. A dominating set of a graph G is a subset S ⊆ V ( G ) of vertices such that every vertex of G is either in S or is adjacent to a vertex in S by an edge in E ( G ) . The domination number of G, denoted γ ( G ) , is the minimal cardinality of a dominating set of G. That is,
γ ( G ) = min { | S | : S is a dominating set of G } . (1.1)
The concept of graph domination was first introduced by Claude Berge in the 1950’s and 1960’s [
Among the many variants of the graph domination problem, one important variant is distance domination. Distance domination allows a vertex in the dominating set to “dominate” not only the vertices directly adjacent to it, but it also allows for that vertex to “dominate” all vertices within a certain distance of it. More precisely, a distance-k dominating set of a graph G is a set S ⊆ V ( G ) of vertices such that every vertex v ∈ V ( G ) is either in S or there exists a vertex s ∈ S such that the distance between v and s is at most k. In other words, S is a distance-k dominating set of G if and only if we can reach any vertex w in G by following a path of length at most k which starts on a vertex in S and terminates at w. Note that the length of a path is just the number of edges in that path. Similar to the standard domination number of a graph, the distance-k domination number of a graph G is the minimal cardinality of a distance-k dominating set of G. We remark that distance-k domination is a generalization of standard domination, as the definition of standard domination is equivalent to that of distance-1 domination.
The concept of distance domination was first motivated in 1991 by Henning, Oellermann, and Swart [
For positive integer parameters t and r, a further generalization of a graph’s domination number is known as the (t, r) broadcast domination number, first defined by Blessing, Insko, Johnson, and Mauretour in 2014 [
The (t, r) broadcast domination parameter, first studied by Blessing et al. in [
In this paper, we expand on the study of domination by introducing a new domination variant which we henceforth refer to as directed (t, r) broadcast domination. Colloquially, the directed (t, r) broadcast domination number is found by taking a directed graph and repeating the standard (t, r) broadcast domination process, but we only allow signal to travel in the same direction as each of the graph’s arcs. Note that this means that signal cannot travel “both ways” as it was previously able to over an edge in the undirected case. Hence, it is necessarily harder for signal to propagate from the towers toward other vertices within the graph. An application of directed (t, r) broadcast domination arises by asking questions of standard (t, r) broadcast domination in networks on which “resources” travel only in a single direction. Such settings include irrigation systems, electrical circuits, and regions of cities which are dense in one-way streets. We emphasize that the introduction of directed (t, r) broadcast domination is a major contribution of this paper, which is the first to define the concept and study results arising from this new graph parameter. Moreover, we remark that all results in this paper provide a new direction for research, thereby opening many new avenues for future work in this area.
OverviewThe remainder of this paper is organized as follows. We begin with Section 2, which provides some important and technical definitions along with the necessary notation to make our understanding of domination theory and (t, r) broadcast domination precise. In Section 3 we investigate the directed (t, r) broadcast domination interval ( D B D I t , r ), which is the smallest interval of integers containing the (t, r) broadcast domination number of each orientation of a graph G. We focus first on the D B D I t , r on arbitrary graphs (Theorems 3.1 and 3.2). Next, we find intervals contained in the D B D I t , r of some small grid graphs G m , n (Proposition 3.10). Additionally, we prove tight bounds on the D B D I t , r for the star graph Sn (Propositions 3.12, 3.13, 3.14) and the fullness of this interval (Theorem 3.15). Then, in Section 4 we introduce the density of directed (t, r) broadcast domination on the infinite grid, focusing on the case ( t , r ) = ( 2 , 2 ) , where we give examples of orientations of the infinite grid which achieve efficient
tower density 1 3 , 1 2 , and 2 3 (Theorem 4.2). The paper concludes with Section
5, which provides numerous directions for further research in directed (t, r) broadcast domination.
In this paper we follow the notation presented in Blessing et al. [
A graph G is an ordered pair ( V ( G ) , E ( G ) ) where V ( G ) is a set of vertices and E ( G ) is a multi-set of edges. An edge is a subset of V ( G ) of size one or two. A directed graph (digraph) is a graph whose edges are ordered pairs of vertices. To differentiate between directed and undirected edges, we henceforth refer to directed edges as arcs. Parallel arcs refers to a set of multiple arcs starting and ending at the same pair of vertices, and a loop is an arc that begins and ends at the same vertex. A simple digraph is a digraph without parallel arcs or loops. A vertex v is incident to an edge e if v ∈ e . Two vertices v 1 , v 2 are adjacent if there exists an edge e such that v 1 , v 2 ∈ e . Lastly, an oriented graph is a simple digraph that does not contain doubly directed arcs, meaning that if (u, v) is the arc from vertex u to v, then there is no arc (v, u) from v to u. In other words, an oriented graph is a simple digraph where each edge is assigned a specified orientation.
We now introduce formally the concept of (undirected) (t, r) broadcast domination following the conventions and notation of Blessing et al. [
Definition 2.1. Throughout we let t ∈ ℕ : = { 1 , 2 , 3 , ⋯ } . Given two vertices u , v ∈ V ( G ) , the distance between u and v, denoted d ( u , v ) , is the minimum length of the paths connecting u and v. We say that v ∈ V ( G ) is a broadcasting vertex of transmission strength t if u transmits a signal of strength t − d ( u , v ) to every vertex u ∈ V ( G ) with d ( u , v ) < t . In the case where d ( u , v ) ≥ t , then the broadcast vertex of transmission strength t does not transmit any signal to the vertex u.
Definition 2.2. Given a vertex v and integer t, the distance t neighborhood of v is defined as N t ( v ) = { w ∈ V ( G ) : d ( w , v ) < t } . If v is selected to be a broadcasting vertex, then we call N t ( v ) the broadcasting neighborhood of v. Given a set of broadcast vertices S ⊆ V ( G ) , each with transmission strength t, we say that the reception at vertex w ∈ V ( G ) is
r ( w ) = ∑ v ∈ S ∩ N t ( w ) ( t − d ( w , v ) ) . (2.1)
That is, the reception r ( w ) is the sum of transmissions from neighboring broadcast vertices in S.
Definition 2.3. A set S ⊆ V ( G ) is called a (t, r) broadcast dominating set if every vertex w ∈ V ( G ) has a reception strength r ( w ) ≥ r . For a finite graph G, the minimal cardinality among all broadcast dominating sets of G is called the (t, r) broadcast domination number of G and is denoted γ t , r ( G ) .
Example 2.4.
Example 2.5. For a given graph, there may be many possible broadcast dominating sets of minimal cardinality.
It is important to note that the dominating sets presented in
Now, we introduce directed (t, r) broadcast domination, the main object of study
in this paper. The following are previous definitions which we extend to an orientation G → of G. That is, we orient all of the edges of G, letting G → be the resulting digraph, and define (t, r) broadcast domination on G → .
Definition 2.6. Let t ∈ ℕ : = { 1,2,3, ⋯ } . Given two vertices u , v ∈ V ( G → ) the distance from u to v, denoted d ( u , v ) , is the minimum length of the directed paths from u to v. We say that u ∈ V ( G → ) is a broadcasting vertex of directed transmission strength t if it transmits a signal of strength t − d ( u , v ) to every vertex v ∈ V ( G ) with d ( u , v ) < t . Once again, vertices which are distance at least t from u receive signal of strength 0 from u.
Note that in directed graphs it is not necessarily the case that d ( u , v ) = d ( v , u ) . Also note that if there is no directed path from u to v, then d ( u , v ) = ∞ . Furthermore, we adopt the following two conventions. Firstly, the transmission strength is omitted when it is clear from context. Secondly, this paper uses the terms “broadcasting vertex” and “tower” interchangeably. Now we continue with other important definitions.
Definition 2.7. Given a vertex v ∈ V ( G → ) and positive integer t, the distance t out-neighborhood of v is defined as N t + ( v ) = { w ∈ V ( G ) : d ( v , w ) < t } . Likewise, the distance t in-neighborhood of v is defined as N t − ( v ) = { w ∈ V ( G ) : d ( w , v ) < t } . If v is selected to be a broadcasting vertex, then we call N t + ( v ) the broadcasting out-neighborhood of v. Given a set of broadcast vertices S ⊆ V ( G → ) , each with transmission strength t, we say that the directed reception at vertex w ∈ V ( G → ) is
r → ( w ) = ∑ v ∈ S ∩ N t − ( w ) ( t − d ( v , w ) ) . (2.2)
That is, the directed reception r → ( w ) is the sum of the out-transmissions from its in-neighboring broadcast vertices in S.
Definition 2.8. A set S ⊆ V ( G → ) is called a directed (t, r) broadcast dominating set if every vertex w ∈ V ( G → ) has a directed reception strength r → ( w ) ≥ r . For a finite digraph G → , the minimal cardinality among all broadcast dominating sets of G → is called the directed (t, r) broadcast domination number of G → and is denoted γ t , r ( G → ) .
For any simple graph G there are 2 | E ( G ) | orientations of the edges of G. Let G be the set of all orientations of the edges of the graph G. Then we are interested in determining possible values of γ t , r ( G → ) whenever G → ∈ G . This motivates the following.
Definition 2.9. Let A t , r ( G ) : = { γ t , r ( G → ) | G → ∈ G } . The directed (t, r) broadcast domination interval of a graph G, denoted D B D I t , r ( G ) , is the interval [ d , D ] , where d = min ( A t , r ( G ) ) and D = max ( A t , r ( G ) ) .
Colloquially, the directed (t, r) broadcast domination interval of G is smallest contiguous interval of integers which contains all possible values of γ t , r ( G → ) for G → ∈ G . Such a concept is motivated by the case in which signal (or some other resource) may only travel in a single direction, but the user maintains control over that direction. For example, a factory manager may wish to establish a traffic pattern for a factory line in her facility, and she may want to know the discrepancies between different directions of travel throughout the plant. She may also wish to know how drastic such trade-offs could be, for which knowing the width of the D B D I t , r would be useful. (We remark as an aside, that in a similar spirit, directed cliques in directed random graphs have been recently been investigated with applications to directed social networks and citation networks [
This section concerns the directed (t, r) broadcast domination interval, first discussing arbitrary graphs before focusing on stars and small grid graphs.
One interesting question surrounding the directed (t, r) broadcast domination interval is whether the name interval is actually appropriate. Namely, given a graph G and fixed values t and r, does D B D I t , r ( G ) = [ d , D ] contain all integer values within this interval. In other words, for all b ∈ [ d , D ] ∩ ℤ , does there exist an orientation G → such that γ t , r ( G → ) = b ? In this section, we show that in the case r = 1 the answer is yes, and we conjecture this to be true for all r-values. Our first result shows that for a fixed t and for r = 1 , flipping an arc in any orientation of a graph cannot change the directed (t, r) broadcast domination number of the graph by more than 1.
Lemma 3.1. Let t be any positive integer. Given a graph G, let G → 0 and G → 1 be two orientations of G where G → 1 can be obtained from G → 0 by flipping (reversing the direction of) a single arc. Then | γ t ,1 ( G → 0 ) − γ t ,1 ( G → 1 ) | ≤ 1 .
Proof. Without loss of generality, suppose G → 0 is the starting orientation, and let a = ( u , v ) be the arc which is flipped to obtain G → 1 so that a becomes a ′ = ( v , u ) . If S ⊆ V ( G ) is a (t, 1) directed broadcast dominating set of G → 0 with | S | = γ t ,1 ( G → 0 ) , then S ∪ { v } is a (t, 1) directed broadcast dominating set of G → 1 as we have now ensured that v has reception t ≥ 1 since it is now a tower. Moreover, the size of a directed (t, 1) broadcast dominating set has increased by at most 1. Again, note that because v is now a broadcasting vertex, the transmission strength at v has strictly increased if v was not previously a broadcasting vertex, or it has stayed the same if v was already a broadcasting vertex. Thus, the directed reception at all other vertices is at least what it was using the dominating set S. This implies that γ t ,1 ( G → 1 ) ≤ γ t ,1 ( G → 0 ) + 1 . A visual representation of this argument is shown in
arc once again would cause the directed (t, r) broadcast domination number to increase by more than 1, a contradiction, since we assumed | S | was the directed (t, 1) broadcast domination number of the initial orientation G → 0 .
Theorem 3.2. Given a graph G, let D B D I t ,1 ( G ) = [ d , D ] . Then, for every b ∈ ℤ ∩ [ d , D ] , there exists an orientation G → b with γ t ,1 ( G → b ) = b .
Proof. Let D B D I t ,1 ( G ) = [ d , D ] . Let G → d , G → D be the orientations of G which achieve directed (t, 1) broadcast domination numbers d and D, respectively. Notice that G → D can be obtained from G → d by a series of arc flips. As we perform these arc flips, the directed (t, 1) broadcast domination number of the resulting graph can never change by more than one, as we established in Lemma 3.1. This implies that at some point each integer value in ℤ ∩ [ d , D ] was attained.
Interestingly, an identical argument, as that presented in Theorem 3.2, also holds in the case ( t , r ) = ( 2 , 2 ) . This is because signal from any broadcasting vertex reaches only that broadcasting vertex’s immediate out-neighborhood. As a result, we have the following.
Theorem 3.3. Given a graph G, let D B D I 2,2 ( G ) = [ d , D ] . Then, for every b ∈ ℤ ∩ [ d , D ] , there exists an orientation G → b with γ 2,2 ( G → b ) = b .
Proof. The argument is identical to that of Lemma 3.1 and Theorem 3.2. Consider graphs G → 0 and G → 1 as before. Because signal form any broadcasting vertex reaches only the out-neighborhood of that vertex, flipping such an arc causes at most one vertex to lose sufficient reception. To mitigate this, include that vertex in the set of broadcasting vertices for G 1 , and the resulting set forms a directed (2, 2) broadcast dominating set. As a result, γ 2,2 ( G → 1 ) − γ 2,2 ( G → 0 ) ≤ 1 , and by symmetry we have that | γ 2,2 ( G → 0 ) − γ 2,2 ( G → 1 ) | ≤ 1 . This immediately proves the desired result.
It is important to illustrate, however, that the same argument does not hold in general for arbitrary t and r.
Let G → T denote the transpose orientation of G → , which orients every arc in G → in the opposite direction in G → T . The following is another known result from 2013 about digraph parameters. While it appears to be useful in proving the fullness of the directed (t, r) broadcast domination interval for arbitrary t and r, the first condition is often not true for the directed (t, r) broadcast domination number of a graph’s orientation.
Theorem 3.4. (Theorem 2.1, [
1) β ( G → ) = β ( G → T ) .
2) If ( u , v ) ∈ E ( G → ) and G → 0 is obtained from G → by replacing (u, v) by (v, u) (i.e., reversing the orientation of one arc), then | β ( G → ) − β ( G → 0 ) | ≤ 1 .
Then for any two orientations G → 1 and G → 2 of the same graph G, | β ( G → 2 ) − β ( G → 1 ) | ≤ ⌊ | E ( G ) | 2 ⌋ .
For the directed (t, r) broadcast domination parameter, failure of the first condition in Theorem 3.4 is perhaps most easily seen on the star on n vertices, shown in
Moreover, for some graphs and given some positive integers t and r, flipping an arc may change the (t, r) broadcast domination number by more than 1, making Lemma 3.1 not possible to generalize for arbitrary t and r. An example of an instance where an arc flip changes the directed (5, 3) broadcast domination number of a graph from two to four can be seen in
In this section, we turn our attention to the directed (t, r) broadcast domination interval on small grid graphs, a family of graphs discussed in [
Before defining the grid graph, we recall that the Cartesian product of graphs G and H, denoted G □ H , has vertex set V ( G ) × V ( H ) , and it has edge set
E ( G □ H ) = { { ( u 1 , v 1 ) , ( u 2 , v 2 ) } such that { u 1 = u 2 and { v 1 , v 2 } ∈ E ( H ) ; or v 1 = v 2 and { u 1 , u 2 } ∈ E ( G ) } } . (3.1)
Definition 3.5. For positive integers m, n, the grid graph with m rows and n columns, denoted G m , n , is the graph obtained by taking the Cartesian product P m □ P n , where P i denotes the path graph on i vertices.
Now, we state a key observation about the directed (t, r) broadcast domination number.
Observation 3.6. Given a graph G, let G → be an arbitrary orientation of G. For positive integers t and r, γ t , r ( G ) ≤ γ t , r ( G → ) .
Observation 3.6 follows from the fact that a (t, r) broadcast dominating set of G → is also necessarily a (t, r) broadcast dominating set of G, as G can be thought of as the result of replacing every singly-directed edge in G → with a doubly-directed one, allowing reception to flow more freely within the undirected graph. Given this observation, a natural question to ask is when this inequality becomes an equality. In other words, does there always exist an orientation G → of a graph G for which γ t , r ( G ) = γ t , r ( G → ) ?
To answer this question, we first highlight some known results for undirected grid graphs. In 2014, [
Theorem 3.7. (Theorems 2.1-2.5, [
1) For n ≥ 3 , the (2, 2) broadcast domination number of the 3 × n grid is
γ 2 , 2 ( G 3 × n ) = ⌈ 4 n 3 ⌉ . (3.2)
2) For n ≥ 4 , the (2, 2) broadcast domination number of the 4 × n grid is
γ 2,2 ( G 4 × n ) = 2 n − ⌈ n − 6 4 ⌉ . (3.3)
3) For n ≥ 5 , the (2, 2) broadcast domination number of the 5 × n grid is
γ 2 , 2 ( G 5 × n ) = 2 n + ⌈ n + 2 7 ⌉ (3.4)
4) For n ≥ 3 , the (3, 1) broadcast domination number of the 3 × n grid is
γ 3,1 ( G 3 × n ) = ⌈ n 3 ⌉ . (3.5)
5) For n ≥ 4 , the (3, 1) broadcast domination number of the 4 × n grid is
γ 3,1 ( G 4 × n ) = ⌊ n + 1 7 ⌋ + ⌊ n + 3 7 ⌋ + ⌊ n + 5 7 ⌋ + 1. (3.6)
While Blessing et al. [
Observation 3.8. In the case ( t , r ) = ( 2 , 2 ) , the process of directing edges is simple: make all broadcasting vertices sources, thereby making all non-broadcasting vertices sinks. Since each vertex in G is either a broadcasting vertex or a neighbor of at least two broadcasting vertices, it suffices to observe that, for every non-broadcasting vertex v in V ( G ) , its resulting in-neighborhood in the directed graph G → remains its previous neighborhood in the undirected graph G. Similarly, for every broadcasting vertex, its resulting out-neighborhood is exactly its neighborhood in the undirected graph G. Thus, each vertex gets the same reception as it previously did in the undirected graph.
Observation 3.9. In the case ( t , r ) = ( 3 , 1 ) , the intuition behind orienting the graph is to direct edges such that signal can travel as far outward from the broadcasting vertices as possible. We again begin by making all broadcasting vertices sources. Then, for each vertex which is an out-neighbor of a broadcasting vertex, direct the remainder of its not-already-directed edges outward. All other edges are incident to two non-broadcasting vertices which are each at a distance 2 away from any broadcasting vertex. This means that each of these non-broadcasting vertices receives signal 1 from that broadcasting vertex and cannot propagate signal further. As a result, this last set of edges can be oriented arbitrarily to complete the construction of a directed grid graph which maintains the (3, 1) dominating set. We are guaranteed that this set must in fact form a (3, 1) dominating set because, for any broadcasting vertex v, the broadcasting out-neighborhood of v in the oriented graph is exactly the broadcasting neighborhood of v in the undirected graph.
Now, for m ∈ { 2,3,4 } we provide intervals of integers which are properly contained in the D B D I 2,2 ( G m , n ) . Note that since we have shown in the previous section that the (2, 2) and (3, 1) directed broadcast domination intervals of a graph must be full, finding an orientation which achieves a domination number greater than the lower bound immediately implies the existence of orientations which achieve all values in between as well. For each of the following small grid graphs, the upper bound is achieved by an orientation of G m , n which achieves the maximum possible number of vertices with in-degree 1. Note also that the OEIS sequence A000111 counts half the number of alternating permutations on n letters. We let A000111(n) denote the nth term of this sequence. We can now state the next result.
Proposition 3.10. Let n ≥ 1 . Then
1) D B D I 2,2 ( G 2, n ) ⊇ [ n , ⌊ 3 n + 2 2 ⌋ ] .
2) D B D I 2,2 ( G 3, n ) ⊇ [ ⌈ 4 n 3 ⌉ , ⌊ 3 n + 2 2 ⌋ + ⌊ 3 n + 4 4 ⌋ ] .
3) D B D I 2,2 ( G 4, n ) ⊇ [ 2 n − ⌈ n − 6 4 ⌉ , ⌊ 3 n + 2 2 ⌋ + ⌊ 3 n + 4 4 ⌋ + k n ] , where
k n = ⌊ A000111 ( n + 1 ) / A000111 ( n ) ⌋ , (3.7)
which represents the number of towers in the fourth row with n columns.
Proof. As previously stated, the orientation of G m , n which achieves the minimal value within its respective interval is achieved by orienting the graph to preserve the dominating sets stated in [
∑ v ∈ V ( G ) d e g − ( v ) = | V ( G ) | , (3.8)
there must exist a vertex with an impossibly large in-degree for the given equation to still hold.
Unfortunately, showing set equality as opposed to the set containment proven in Proposition 3.10 would require proof that all orientations of these grid graphs achieve (t, r) broadcast domination interval strictly within this interval. In the case of (2, 2) domination, we conjecture that the proof lies in the fact that a vertex must be a dominating vertex if and only if it has in-degree at most 1, implying that the orientations above maximize the number of such vertices achieve the maximal (2, 2) broadcast domination number.
To aid in further understanding the directed (t, r) broadcast domination intervals on small grid graphs, we provide Sage Code [
In the previous section, we found an interval of directed (t, r) broadcast domination numbers which is contained in a (potentially larger) directed (t, r) broadcast domination interval of small grid graphs. In this section, we walk through a full characterization of the directed (t, r) broadcast domination interval for the star, a common and relatively simple-to-understand family of graphs. We begin by formally defining a star below.
Definition 3.11. A star on n vertices, denoted Sn, consists of a single central vertex to which n − 1 leaf vertices are adjacent.
In general domination theory, the star is a simple example of a graph with domination number 1, as the central vertex is adjacent to all other vertices in the graph. Additionally, characterizing the undirected (t, r) broadcast domination number of a star is also quite easy. Specifically, if t > r , then γ t , r ( S n ) = 1 . Otherwise, γ t , r ( S n ) = n . However, once we consider all orientations of Sn, we see that characterizing the directed (t, r) broadcast domination interval of a star becomes noticeably more involved. What follows is a full classification of D B D I t , r ( S n ) for n ≥ 3 .
Proposition 3.12. Let Sn be the star graph on n vertices with n ≥ 3 , and let t, r be integers such that t = r . If t = r = 1 , then γ t , r ( S → n ) = n for all orientations S → n of Sn. If t = r = 2 , then D B D I t , r ( S n ) = [ n − 1 , n ] .
Proof. The case t = r = 1 is trivial because all vertices must be broadcasting vertices regardless of the graph orientation. If t = r = 2 , notice in
Proposition 3.13. Let Sn be the star graph on n vertices with n ≥ 3 , and let t, r be integers. If t = r > 2 , then D B D I t , r ( S n ) = [ 2 , n ] .
Proof. Let s denote the number of source leaves in an orientation of Sn, and let S n s denote the orientation of the star with s source leaves. To show that D B D I t , r ( S n ) = [ 2 , n ] , we start with S n 0 , the leftmost graph in
Proposition 3.14. Let Sn be the star graph on n vertices with n ≥ 3 , and let t, r be integers. If t > r , then D B D I t , r ( S n ) = [ 1 , n − 1 ] .
Proof. Once again, let s denote the number of source leaves in an orientation of Sn, and let S n s denote the orientation of the star with s source leaves. To show that D B D I t , r ( S n ) = [ 1 , n − 1 ] , we start with S n 0 . The central vertex is required to be a dominating vertex because it has in-degree zero, and all leaves receive sufficient reception from this central vertex, making the domination number of this orientation 1. We now proceed to flip the orientation of each edge so that s increases from 0 to n − 1. If ( t , r ) = ( 2 , 1 ) , each of these flips increases the domination number by 1 except for the last flip, when s increases from n − 2 to n − 1. This is because the set of broadcasting vertices gains the last leaf vertex but loses the central vertex when this arc is flipped. This case is highlighted in
We remark that this result leads to a noteworthy result about the D B D I t , r ( S n ) .
Theorem 3.15. Let t , r , d , D be positive integers such that [ d , D ] = D B D I t , r ( S n ) and t ≥ r . Then, for every x ∈ [ d , D ] , there exists an orientation S → n x such that γ t , r ( S → n x ) = x . In other words, the directed (t, r) broadcast domination interval of a star on n vertices is always full.
Proof. By exhaustion, using the previous propositions within this section.
As exhibited by the star graph, finding the directed (t, r) broadcast domination interval can be very difficult, even for simple families of graphs. Moreover, even if finding a dominating “strategy” may be straightforward, proving that the interval resulting from that “strategy” is equal to that graph’s (t, r) directed broadcast domination interval can be quite difficult, as exhibited by our findings from this section and from Section 3.2.
We now shift our discussion to (t, r) broadcast domination on the infinite grid graph, denoted by ℤ × ℤ . In previous literature, Blessing et al. and Harris et al. discuss efficient domination of the infinite Cartesian and triangular lattices, respectively [
Definition 4.1. ( [
r ( u ) = ( r if d ( u , v ) ≥ t − r for all v ∈ S r − d ( u , v ) if 0 ≤ d ( u , v ) < t − r for exactly one v ∈ S . (4.1)
In other words, non-broadcasting vertices far away from broadcasting vertices receive only the minimum required signal and do not “waste” signal by receiving more than they need. Of course, vertices which are relatively close to broadcasting vertices are not penalized for having more than the minimum required reception because they must continue transmitting signal to vertices which are farther away.
Now equipped with a notion of efficiency, we present our main result.
Theorem 4.2. There exist orientations of the graph ℤ × ℤ that achieve efficient directed (2, 2) broadcast domination densities of 1 3 , 1 2 , and 2 3 .
Proof. The proof is via construction. To give each of these results, consider the orientations in
horizontal line are 1 3 and 2 3 of the total number of vertices, respectively.
Motivated by Theorem 4.2 and our prior study of the directed (t, r) broadcast domination interval, we pose the following open problems for further study.
Question 4.3. Does every rational number d ∈ [ 1 3 , 2 3 ] appear as a (2, 2)
directed broadcast domination density? If not, classify the rational numbers in that interval that do appear.
Question 4.4. Is there a (2, 2) directed broadcast domination density d such that the density d is irrational, i.e., d ∈ [ 1 3 , 2 3 ] ∩ ℚ c ?
One may also ponder the implications of periodic tower placement on various topological surfaces.
Question 4.5. Do the three (t, r) directed broadcast dominating sets given in
We provide a partially negative answer to Question 4.5 by noting that the infinite grid achieving tower density 1 3 in
aperiodic orientation (by orienting the missing arcs accordingly), but this orientation always has rational tower density. We know that regardless of the orientation of the missing arcs, the resulting (t, r) broadcast dominating set remains efficient because those missing arcs carry no signal. Of course, this infinite grid may also be oriented periodically to achieve the same effect, and we note that one reason for this flexibility in orienting the arcs is that no signal travels along these arcs. In other words, for each of these arcs (u, v) left blank in
Still, it is the case that all three of these domination densities are realizable as dominating sets on the same toroidal grid, and it is important to note that this fact might help us determine which rational numbers are possible dominating densities. Exploring this further may provide connections to topology and also provide an answer to Question 4.3.
Interestingly, the orientations of the infinite grid depicted in
Proposition 4.6. Given integers m , n ≥ 6 ,
D B D I 2,2 ( G m , n ) ⊇ ( [ d , ⌊ 2 m n 3 ⌋ ] if n mod 3 ≡ 0 [ d , ⌊ 2 m ( n − 1 ) 3 ⌋ + m ] if n mod 3 ≡ 1 [ d , ⌊ 2 m ( n − 2 ) 3 ⌋ + m + ⌈ m 2 ⌉ ] if n mod 3 ≡ 2 (4.2)
where the lower bound
d = ( ⌈ ( m + 2 ) ( n + 2 ) 3 ⌉ − 6 if m ≡ n mod 3 ⌈ ( m + 2 ) ( n + 2 ) 3 ⌉ − 5 if m ≡ n mod 3 (4.3)
is the best-known upper bound on γ ( 2,2 ) ( G m , n ) previously established by Blessing et al. ( [
Proof. To achieve the upper bound of this interval, orient the grid as in
To count the number of red vertices in the rectangular region, we let m, n denote the number of rows and columns in the rectangular region, respectively. Then we consider three separate cases when n mod 3 is 0, 1, or 2. In the first case, one third of the rows will have only red vertices, while the two to the right of it will have equally many red and non-red vertices. Thus the overall density of
red vertices is ⌊ 2 m n 3 ⌋ . In the second case, we break this grid into two sub-grids:
the first being a grid with n − 1 columns which falls into the previous case, and the second being a grid with 1 column and all red vertices. Thus the overall
number of red vertices is ⌊ 2 m ( n − 1 ) 3 ⌋ + m . In the third case, we follow a similar
pattern, again breaking the grid into a sub-grid with n − 2 columns which falls into the first case, then a column of all red vertices, and then a column with every other vertex red. This grid achieves an overall red vertex density of
⌊ 2 m ( n − 2 ) 3 ⌋ + m + ⌈ m 2 ⌉ .
We first claim that the set of towers within this region on the infinite graph also forms a dominating set of G → m , n . Furthermore, we claim that this set is minimal, thereby achieving the domination number of G → m , n .
To establish the first claim, notice that all vertices within this region with in-degree 0 or 1 are dominating vertices because having in-degree 0 or 1 implies that such a vertex can receive reception of at most 1 from a neighbor. All other vertices have in-degree at least 2 in G → m , n are non-dominating vertices. This implies that the set of dominating vertices in G → m , n forms a dominating set of the graph.
The second claim is established by the fact that every vertex with in-degree 0 or 1 must be a dominating vertex, as it cannot receive sufficient reception from only a single other vertex. Additionally, any vertex with in-degree greater than 1 is not a dominating vertex. Thus, this set of dominating vertices is a minimal (2, 2) broadcast dominating set.
The lower bound of this interval was proven by Blessing et al. for undirected grids, and it still holds when we direct all edges away from the broadcasting towers as in the left image in
It is also important to note that, much like in the case of finite graphs, there exist infinite grid graphs that cannot be oriented to preserve their (t, r) broadcast dominating sets. One example is on the infinite triangular grid, which cannot be oriented to preserve the (4, 3) dominating set given by Harris et al. in [
Example 4.7. As pictured in
lattice to preserve the dominating set given by Harris et al. in [
In Section 3, we prove that the directed (t, r) broadcast domination interval is full when t ≥ r = 1 and in the isolated case ( t , r ) = ( 2 , 2 ) . We conjecture that this is true for arbitrary t ≥ r ≥ 1 . One direction of study is proving or providing a counterexample to the following.
Conjecture 5.1. Let t and r be positive integers such that t ≥ r . Given a graph G, let D B D I t , r ( G ) = [ d , D ] , where
d = min { γ t , r ( G → ) : G → is an orientation of G } (5.1)
and
D = max { γ t , r ( G → ) : G → is an orientation of G } . (5.2)
For every b ∈ ℤ ∩ [ d , D ] , there exists an orientation G → b with γ t , r ( G → b ) = b .
We also study directed (t, r) broadcast domination on small grid graphs. There are many possible directions for further study, chief of which would be finding a closed formula for the directed (t, r) broadcast domination interval of an arbitrary finite grid graph.
Research Project 5.2. Let t , r , m , n be positive integers, and let t ≥ r ≥ 1 . Find a formula for D B D I t , r ( G m , n ) as a function of t , r , m and n.
Lastly in Section 3, we study the directed (t, r) broadcast domination interval for the star graph. In light of those results, one could study the directed (t, r) broadcast domination interval of other graph families.
Research Project 5.3. Let t and r be positive integers such that t ≥ r ≥ 1 . For a family of graphs, find the D B D I t , r of that graph family as a function of t, r, and the number of vertices in the graph. Examples of interesting graphs to consider include cycles, tournaments (directed complete graphs), and spiders (a collection of paths which all share one endpoint).
In Section 4, we introduce directed (t, r) broadcast domination on the infinite grid, and we show an orientation of the infinite grid which efficiently achieves
broadcasting vertex densities 1 3 , 1 2 , and 2 3 when ( t , r ) = ( 2 , 2 ) . One future direction of study concerns answering the following question.
Question 5.4. Let ( t , r ) = ( 2 , 2 ) . For d ∈ ℚ ∩ [ 1 3 , 2 3 ] , does there exist an orientation of the infinite Cartesian grid which achieves tower density d? If not, for which rational numbers d ∈ [ 1 3 , 2 3 ] can densities be achieved?
There are many other ways to study directed (t, r) broadcast domination on the infinite grid. One potential direction could be to explore other known efficient (t, r) broadcast dominating patterns (i.e., those studied in [
P. E. Harris acknowledges funding support from The Karen EDGE Fellowship Program and P. Hollander thanks Williams College Science Center for funding in support of this research.
The authors declare no conflicts of interest regarding the publication of this paper.
Harris, P.E., Hollander, P. and Insko, E. (2022) On (t, r) Broadcast Domination of Directed Graphs. Open Journal of Discrete Mathematics, 12, 78-100. https://doi.org/10.4236/ojdm.2022.123006