aa r X i v : . [ m a t h . C O ] M a y Broadcast Dimension of Graphs
Jesse Geneson and Eunjeong Yi Iowa State University, Ames, IA 50011, USA Texas A&M University at Galveston, Galveston, TX 77553, USA [email protected] ; [email protected] May 18, 2020
Abstract
In this paper we initiate the study of broadcast dimension, a variant of metricdimension. Let G be a graph with vertex set V ( G ), and let d ( u, w ) denote the lengthof a u − w geodesic in G . For k ≥
1, let d k ( x, y ) = min { d ( x, y ) , k + 1 } . A function f : V ( G ) → Z + ∪ { } is called a resolving broadcast of G if, for any distinct x, y ∈ V ( G ),there exists a vertex z ∈ V ( G ) such that f ( z ) = i > d i ( x, z ) = d i ( y, z ). The broadcast dimension , bdim( G ), of G is the minimum of c f ( G ) = P v ∈ V ( G ) f ( v ) over allresolving broadcasts of G , where c f ( G ) can be viewed as the total cost of the transmitters(of various strength) used in resolving the entire network described by the graph G .Note that bdim( G ) reduces to adim( G ) (the adjacency dimension of G , introduced byJannesari and Omoomi in 2012) if the codomain of resolving broadcasts is restricted to { , } . We determine its value for cycles, paths, and other families of graphs. We provethat bdim( G ) = Ω(log n ) for all graphs G of order n , and that the result is sharp upto a constant factor. We show that adim( G )bdim( G ) and bdim( G )dim( G ) can both be arbitrarily large,where dim( G ) denotes the metric dimension of G . We also examine the effect of vertexdeletion on the adjacency dimension and the broadcast dimension of graphs. Keywords: metric dimension, adjacency dimension, resolving broadcast, broadcast dimension
Let G be a finite, simple, and undirected graph with vertex set V ( G ) and edge set E ( G ). The distance between two vertices x, y ∈ V ( G ), denoted by d ( x, y ), is the length of a shortest pathbetween x and y in G ; if x and y belong to different components of G , we define d ( x, y ) = ∞ .Metric dimension, introduced by Slater [24] and by Harary and Melter [13], is a graph parameterthat has been studied extensively. A vertex z ∈ V ( G ) resolves a pair of vertices x, y ∈ V ( G ) if d ( x, z ) = d ( y, z ). A set S ⊆ V ( G ) is a resolving set of G if, for any distinct x, y ∈ V ( G ), there exists z ∈ S such that d ( x, z ) = d ( y, z ). The metric dimension of G , denoted by dim( G ), is the minimumcardinality over all resolving sets of G . Khuller et al. [19] considered robot navigation as one of theapplications of metric dimension, where a robot that moves from node to node knows its distancesto all landmarks.For x ∈ V ( G ) and S ⊆ V ( G ), let d ( x, S ) = min { d ( x, y ) : y ∈ S } . Meir and Moon [20] introduceddistance- k domination. For a positive integer k , a set D ⊆ V ( G ) is called a distance- k dominatingset of G if, for each u ∈ V ( G ) − D , d ( u, D ) ≤ k . The distance- k domination number of G , denotedby γ k ( G ), is the minimum cardinality over all distance- k dominating sets of G ; the distance-1domination number is the well-known domination number. Erwin [7, 8] introduced the concept of roadcast domination, where cities with broadcast stations have transmission power that enable themto broadcast messages to cities at distances greater than one, depending on the transmission power ofbroadcast stations. More explicitly, following [7, 8], a function f : V ( G ) → { , , , . . . , diam( G ) } iscalled a dominating broadcast of G if, for each vertex x ∈ V ( G ), there exists a vertex y ∈ V ( G ) suchthat f ( y ) > d ( x, y ) ≤ f ( y ). The broadcast (domination) number , γ b ( G ), of G is the minimumof D f ( G ) := P v ∈ V ( G ) f ( v ) over all dominating broadcasts f of G ; here, D f ( G ) can be viewed asthe total cost of the transmitters used to achieve full coverage of a network of cities described viathe graph G being considered. Note that γ b ( G ) reduces to k · γ k ( G ) if the codomain of dominatingbroadcasts is restricted to { , k } . It is known that determining the domination number of a generalgraph is an NP-hard problem (see [11]).Recently, Jannesari and Omoomi [17] introduced adjacency dimension of G , denoted by adim( G ),as a tool to study the metric dimension of lexicographic product graphs; they defined the adjacencydistance between two vertices x, y ∈ V ( G ) to be 0 , ,
2, respectively, if d ( x, y ) = 0, d ( x, y ) = 1,and d ( x, y ) ≥
2. Adjacency resolving set and adjacency dimension are defined analogously in [17].Assuming that a landmark that can detect long distance can be costly, the authors of [17] considereda robot that detects its position only from landmarks adjacent to it; this can be viewed as combiningthe concept of a resolving set and a dominating set. More generally, we can apply the concept of adistance- k dominating set to a resolving set. If a robot can detect up to distance k > k dimension of G , denoted by dim k ( G ); note that dim ( G ) = adim( G ).Now, we apply the concept of a dominating broadcast to a resolving set. For a positive integer k and for x, y ∈ V ( G ), let d k ( x, y ) = min { d ( x, y ) , k + 1 } . Let f : V ( G ) → Z + ∪ { } be a function.We define supp G ( f ) = { v ∈ V ( G ) : f ( v ) > } . We say that f is a resolving broadcast of G if, forany distinct x, y ∈ V ( G ), there exists a vertex z ∈ supp G ( f ) such that d f ( z ) ( x, z ) = d f ( z ) ( y, z ). The broadcast dimension of G , denoted by bdim( G ), is the minimum of c f ( G ) = P v ∈ V ( G ) f ( v ) over allresolving broadcasts f of G , where c f ( G ) can be viewed as the total cost of the transmitters (ofvarious strength) used in resolving the entire network described via the graph G being considered.Note that, if the codomain of resolving broadcasts is restricted to { , k } , where k is a positiveinteger, then bdim( G ) reduces to k · dim k ( G ). For an ordered set S = { u , u , . . . , u k } ⊆ V ( G )of distinct vertices, the metric code, the adjacency code, and the broadcast code, respectively, of v ∈ V ( G ) with respect to S are the k -vectors r S ( v ) = ( d ( v, u ) , d ( v, u ) , . . . , d ( v, u k )), a S ( v ) =( d ( v, u ) , d ( v, u ) , . . . , d ( v, u k )), and b S ( v ) = ( d i ( v, u ) , d i ( v, u ) , . . . , d i k ( v, u k )), where f ( u j ) = i j > f being considered. It is known that determining the metricdimension (adjacency dimension, respectively) of a graph is an NP-hard problem; see [11] ([10],respectively).Suppose f ( x ) and g ( x ) are two functions defined on some subset of real numbers. We write f ( x ) = O ( g ( x )) if there exist positive constants N and C such that | f ( x ) | ≤ C | g ( x ) | for all x > N , f ( x ) = Ω( g ( x )) if g ( x ) = O ( f ( x )), and f ( x ) = Θ( g ( x )) if f ( x ) = O ( g ( x )) and f ( x ) = Ω( g ( x )).In this paper, we initiate the study of broadcast dimension. In Section 2, we discuss some generalresults on the metric dimension, the adjacency dimension, and the broadcast dimension of graphs.For example, it is easy to see that for any graph G , dim( G ) ≤ bdim( G ) ≤ adim( G ). We also findthe broadcast dimension of paths and cycles. In Section 3, we prove that bdim( G ) = Ω(log n ) forall graphs G of order n , and that the result is sharp up to a constant factor. We also characterizethe family of graphs of adjacency dimension k for each k . In Section 4, we characterize the graphs G such that bdim( G ) equals 1, 2, and | V ( G ) | −
1. It is noteworthy that bdim( G ) = 2 (adim( G ) = 2,respectively) implies that G is planar, whereas an example of non-planar graph G with dim( G ) = 2was given in [19]. In Section 5, we provide graphs G such that both adim( G ) − bdim( G ) andbdim( G ) − dim( G ) can be arbitrarily large. We also show that, for two connected graphs G and H with H ⊂ G , dim( H ) − dim( G ) (bdim( H ) − bdim( G ) and adim( H ) − adim( G ), respectively) canbe arbitrarily large. In addition, we find all trees T such that bdim( T ) = dim( T ). In Section 6,we examine the effect of vertex deletion on adjacency dimension and broadcast dimension. We alsoinvestigate the effect of edge deletion on adjacency dimension. In Section 7, we conclude with some pen problems.We conclude the introduction with some terminology and notation that we will use throughoutthe paper. The diameter , diam( G ), of G is max { d ( x, y ) : x, y ∈ V ( G ) } . The open neighborhood of avertex v ∈ V ( G ) is N ( v ) = { u ∈ V ( G ) : uv ∈ E ( G ) } and its closed neighborhood is N [ v ] = N ( v ) ∪{ v } .The degree of a vertex u in G , denoted by deg( u ), is | N ( u ) | . An end vertex is a vertex of degree one,and a major vertex is a vertex of degree at least three. The join of two graphs H and H , denotedby H + H , is the graph obtained from the disjoint union of two graphs H and H by joining everyvertex of H with every vertex of H . We denote by P n , C n , K n , and K m,n respectively the path,cycle, and complete graph on n vertices, and the complete bipartite graph with parts of size m and n . We denote by α and α , respectively, the α -vector with 1 on each entry and the α -vector with2 on each entry. In this section, we discuss some general results on the metric dimension, the adjacency dimension,and the broadcast dimension of graphs. We also determine the broadcast dimension of paths andcycles. For distinct u, w ∈ V ( G ), if N ( u ) − { w } = N ( w ) − { u } , then u and w are called twin vertices of G . Observation 2.1.
Let u and w be twin vertices of a graph G . Then(a) [16] for any resolving set S of G , S ∩ { u, w } 6 = ∅ ;(b) [17] for any adjacency resolving set A of G , A ∩ { u, w } 6 = ∅ ;(c) for any resolving broadcast f of G , f ( u ) > or f ( w ) > . Proposition 2.2. [17] (a) If G is a connected graph, then adim( G ) ≥ dim( G ) .(b) If G is a connected graph with diam( G ) = 2 , then adim( G ) = dim( G ) . Moreover, there existsa graph G such that adim( G ) = dim( G ) and diam( G ) > .(c) For every graph G , adim( G ) = adim( G ) , where G denotes the complement of G . Observation 2.3. (a) For any graph G of order n ≥ , ≤ dim( G ) ≤ bdim( G ) ≤ adim( G ) ≤ n − .(b) For any graph G with diam( G ) ∈ { , } , dim( G ) = bdim( G ) = adim( G ) . Next, we consider graphs G with diam( G ) ≤
2. For two graphs H and H , diam( H + H ) ≤ H + H ) = bdim( H + H ) = adim( H + H ) by Observation 2.3(b). Theorem 2.4. [1, 23]
For n ≥ , dim( C n + K ) = (cid:26) if n ∈ { , } , ⌊ n +25 ⌋ otherwise. Theorem 2.5. [2]
For n ≥ , dim( P n + K ) = if n = 1 , if n ∈ { , } , if n = 6 , ⌊ n +25 ⌋ otherwise. Proposition 2.2(b), along with Theorems 2.4 and 2.5, implies the following proposition.
Proposition 2.6. [17]
For n ≥ , if G ∈ { P n , C n } , then adim( G + K ) = ⌊ n +25 ⌋ . s an immediate consequence of Observation 2.3(b) and Theorems 2.4 and 2.5, we have thefollowing corollary. Corollary 2.7.
For n ≥ , let G ∈ { P n , C n } . Then bdim( G + K ) = if n = 3 and G = P , if n = 3 and G = C , if n = 6 , ⌊ n +25 ⌋ otherwise. The metric dimension and the adjacency dimension, respectively, of a complete k-partite graphswas determined in [22] and [17].
Theorem 2.8. [17, 22]
For k ≥ , let G = K a ,a ,...,a k be a complete k -partite graph of order n = P ki =1 a i . Let s be the number of partite sets of G consisting of exactly one element. Then dim( G ) = adim( G ) = (cid:26) n − k if s = 0 ,n + s − k − if s = 0 . As an immediate consequence of Observation 2.3(b) and Theorem 2.8, we have the followingcorollary.
Corollary 2.9.
For k ≥ , let G = K a ,a ,...,a k be a complete k -partite graph of order n = P ki =1 a i .Let s be the number of partite sets of G consisting of exactly one element. Then bdim( G ) = (cid:26) n − k if s = 0 ,n + s − k − if s = 0 . Now, we recall the metric dimension of the Petersen graph.
Theorem 2.10. [18]
For the Petersen graph P , dim( P ) = 3 . Since diam( P ) = 2, Observation 2.3(b) and Theorem 2.10 imply the following corollary. Corollary 2.11.
For the Petersen graph P , bdim( P ) = adim( P ) = 3 . Next, we consider paths and cycles.
Proposition 2.12. [17]
For n ≥ , adim( P n ) = adim( C n ) = ⌊ n +25 ⌋ . Theorem 2.13.
For n ≥ , bdim( P n ) = bdim( C n ) = ⌊ n +25 ⌋ Proof.
Let G be P n or C n , with vertices v , . . . , v n − in order, where n ≥
4. By Observation 2.3(a)and Proposition 2.12, bdim( G ) ≤ ⌊ n +25 ⌋ for n ≥
4. Thus it suffices to prove that P v ∈ V ( G ) f ( v ) isminimized when f ( v ) ≤ v ∈ V ( G ).Suppose that f is a resolving broadcast that achieves bdim( G ). If f ( v ) ≤ v ∈ V ( G ),then we are done. Otherwise, we modify f to obtain a new resolving broadcast f ′ for which P v ∈ V ( G ) f ′ ( v ) ≤ P v ∈ V ( G ) f ( v ) and f ′ ( v ) ≤ v ∈ V ( G ).Start by defining f such that f ( v ) = f ( v ) for all v ∈ V ( G ). Given f i , let v j be any vertex in V ( G ) such that f i ( v j ) >
1. If f i ( v j ) = 2, then we define f i +1 ( v ( j −
1) mod n ) = f i +1 ( v ( j +1) mod n ) = 1and f i +1 ( v j ) = 0, unless v j is an end vertex of P n . If G = P n and f i ( v ) = 2, then we define f i +1 ( v ) = 1 and f i +1 ( v ) = 1. If G = P n and f i ( v n − ) = 2, then we define f i +1 ( v n − ) = 1 and f i +1 ( v n − ) = 1.Otherwise if f i ( v j ) = x >
2, then we define f i +1 ( v j ) = x − f i +1 ( v ( j − x +1) mod n ) = f i +1 ( v ( j + x −
1) mod n ) = 1. If any vertices are assigned multiple values for f i +1 , only the maximumvalue is used. If any vertex v is assigned no values for f i +1 , then f i +1 ( v ) = f i ( v ).The process will end in finitely many steps, so suppose that k is an integer such that f k ( v ) ≤ v ∈ V ( G ). Then we let f ′ = f k , and P v ∈ V ( G ) f ′ ( v ) ≤ P v ∈ V ( G ) f ( v ) by construction. Thusbdim( P n ) = adim( P n ) = bdim( C n ) = adim( C n ) = ⌊ n +25 ⌋ . Extremal bounds and characterization
In this section, we prove that bdim( G ) = Ω(log n ) for all graphs G of order n , and that the result issharp up to a constant factor. We also obtain bounds for the clique number and maximum degreeof graphs with adjacency dimension k or broadcast dimension k . Furthermore, we characterize thefamily of graphs of adjacency dimension k . First, we recall some known bounds for the metricdimension of graphs. Theorem 3.1. [4]
For a connected graph G of order n ≥ and diameter d , f ( n, d ) ≤ dim( G ) ≤ n − d, where f ( n, d ) is the least positive integer k for which k + d k ≥ n . Hernando et al. [16] improved the bound in Theorem 3.1.
Theorem 3.2. [16]
Let G be a connected graph of order n , diameter d , and dim( G ) = k . Then n ≤ (cid:18)(cid:22) d (cid:23) + 1 (cid:19) k + k ⌈ d ⌉ X i =1 (2 i − k − . As a corollary of Observation 2.3(a) and Theorem 3.2, we obtain bounds on the maximum orderof any graph G with diam( G ) = d and bdim( G ) = k . Corollary 3.3.
For any graph G with diam( G ) = d and bdim( G ) = k , | V ( G ) | ≤ (cid:18)(cid:22) d (cid:23) + 1 (cid:19) k + k ⌈ d ⌉ X i =1 (2 i − k − . Proof. If G has bdim( G ) = k , then dim( G ) ≤ k by Observation 2.3(a). So, the desired result followsfrom Theorem 3.2.We also obtain bounds on the maximum order of any subgraph of G with diam( G ) = d andbdim( G ) = k . Theorem 3.4. [12]
For any graph G with dim( G ) = k and any subgraph H of G with diam( H ) = d , | V ( H ) | ≤ ( d + 1) k . Corollary 3.5.
For any graph G with bdim( G ) = k and any subgraph H of G with diam( H ) = d , | V ( H ) | ≤ ( d + 1) k .Proof. If G has bdim( G ) = k , then dim( G ) ≤ k by Observation 2.3(a). So, the desired result followsfrom Theorem 3.4. Remark 3.6.
By Observation 2.3(a), Corollaries 3.3 and 3.5 hold when bdim( G ) = k is replacedby adim( G ) = k . The next result shows that Corollary 3.3 is sharp for d = 2. This result uses a family of graphsfrom [27, 12]. Theorem 3.7.
There exist graphs G of order n with bdim( G ) = O (log n ) .Proof. We construct a graph G of order n = k + 2 k by starting with k vertices v , . . . , v k in a clique,and adding 2 k new vertices { u b } b ∈{ , } k also in a clique labeled with binary strings of length k suchthat u b has an edge with v j if and only if the j th digit of b is 1.Define the resolving broadcast f such that f ( v i ) = 1 for all 1 ≤ i ≤ k and f ( u b ) = 0 for all b ∈ { , } k . Since n = k + 2 k and P v ∈ V ( G ) f ( v ) = k , we have bdim( G ) = O (log n ). For any n notof the form k + 2 k , we can define n ′ to be the least number greater than n that is of the form k + 2 k ,construct G ′ with n ′ vertices as described, and delete any number of vertices u b from G ′ until theremaining graph G has n vertices. v v u u u u u u u u K Figure 1:
A graph G of order n satisfying bdim( G ) = O (log n ); here k = 3 for G described in theproof of Theorem 3.7.Based on the proof of Theorem 3.7, we have the following corollary. Corollary 3.8.
There exist graphs G of order n with adim( G ) = O (log n ) . The construction in Theorem 3.7 can also be used to recursively characterize the graphs G withadim( G ) = k . Given any graph G on k vertices v , . . . , v k and G on 2 k vertices { u b } b ∈{ , } k ,define the graph B ( G , G ) to be obtained by connecting v i and u b if and only if the i th digit of b is 1. Moreover, define B ( G , G ) to be the family of induced subgraphs of B ( G , G ) that containevery vertex in G . Finally, define H = ∅ and for each k > H k to be the family of graphsobtained from taking the union of B ( G , G ) over all graphs G with j vertices v , . . . , v j and G with 2 j vertices { u b } b ∈{ , } j , for each 1 ≤ j ≤ k . Theorem 3.9.
For each k ≥ , the set of graphs G with adim( G ) = k is H k − H k − up toisomorphism.Proof. It suffices to show that the set of graphs G with adim( G ) ≤ k is H k . By construction, everygraph in H k has adim( G ) ≤ k , since the vertices v , . . . , v j are an adjacency resolving set. Thus itsuffices to show that every graph G with adim( G ) ≤ k is in H k . Fix an arbitrary graph G withadim( G ) ≤ k . Let X = { x , . . . , x j } be an adjacency resolving set for G with j ≤ k . Let G be theinduced subgraph of G restricted to X , and let G be the induced subgraph of G restricted to X .Label the vertex v of G as u b with a binary string b so that the i th digit of b is 1 if and only if thereis an edge between v and x i . Note that every vertex gets a unique label, or else X would not be anadjacency resolving set. Let G ′ be any graph on 2 j vertices { u b } b ∈{ , } j such that G ′ | V ( G ) = G .Then G is an induced subgraph of B ( G , G ′ ) that contains every vertex in G , so G is in H k .As a corollary, we obtain an upper bound on the maximum order of a graph of adjacencydimension k . The graph in Theorem 3.7 shows that the bound is sharp. Corollary 3.10.
The maximum order of a graph of adjacency dimension k is k + 2 k . We also obtain a sharp upper bound on the maximum degree of a graph of adjacency dimension k . Corollary 3.11.
The maximum possible degree of any vertex in a graph of adjacency dimension k is k + 2 k − .Proof. The upper bound is immediate from Corollary 3.10, while the upper bound is achieved bythe vertex u k in B ( K k , K k ).In addition, we obtain a sharp upper bound on the clique number of graphs of adjacency dimen-sion k and graphs of broadcast dimension k . Corollary 3.12.
The maximum possible clique number of any graph of adjacency dimension k is k . Similarly, the maximum possible clique number of any graph of broadcast dimension k is k . roof. The upper bound follows from Corollary 3.5. The bound is achieved by the graph G = B ( K k , K k ), which has adim( G ) = bdim( G ) = k .The next result is sharp up to a constant factor, as shown by paths, cycles, and grid graphs. Proposition 3.13.
For graphs G of diameter d , adim( G ) ≥ bdim( G ) ≥ d .Proof. Suppose that G is a graph of diameter d with minimal path v , . . . , v d +1 between two vertices v , v d +1 with distance d , and let f be a resolving broadcast that achieves bdim( G ). Then | supp G ( f ) | +2 P v ∈ V ( G ) f ( v ) ≥ d , which implies that bdim( G ) ≥ d .Thus we have a sharp bound on bdim( G ) up to a constant factor for any graph G with dim( G ) = O (1), where the upper bound follows from the definition of bdim( G ). Theorem 3.14.
For every graph G of diameter d , d ≤ bdim( G ) ≤ dim( G )( d − . Corollary 3.15. If G has diameter d and dim( G ) = O (1) , then bdim( G ) = Θ( d ) . If G has diameter d = O (1) , then bdim( G ) = Θ(dim( G )) . The next result is sharp up to a constant factor by Theorem 3.7.
Theorem 3.16.
For all graphs G of order n , adim( G ) ≥ bdim( G ) = Ω(log n ) .Proof. Suppose that f is a resolving broadcast that achieves bdim( G ), and let y = | supp G ( f ) | . TheΩ(log n ) bound holds if y > ln( n ), so we suppose that y ≤ ln( n ). Since f is a resolving broadcastfor G , we must have y + Q v ∈ supp G ( f ) ( f ( v ) + 1) ≥ n , which implies by the arithmetic-geometric meaninequality that y + P v ∈ V ( G ) f ( v ) ≥ y ( n − y ) /y , or equivalently P v ∈ V ( G ) f ( v ) ≥ y ( n − y ) /y − y .Since y ( n − y ) /y ≥ y ( n ) /y ≥ ye for n sufficiently large, we have P v ∈ V ( G ) f ( v ) ≥ e − e y ( n ) /y .Define g ( y ) = ln( e − e y ( n ) /y ), so g ′ ( y ) = y − ln( n ) y , which has one root at y = ln( n ). Thisis a minimum since g ′ ( y ) < y < ln( n ) and g ′ ( y ) > y > ln( n ). Since ln( x ) is anincreasing function, e − e y ( n ) /y is also minimized at y = ln( n ), where it has value ( e −
1) ln( n ).Thus P v ∈ V ( G ) f ( v ) ≥ ( e −
1) ln( n ) in this case. G having bdim( G ) equal to , , and | V ( G ) | − Next, we characterize graphs G having bdim( G ) equal to 1, 2, and | V ( G ) | −
1. We begin with thefollowing known results on metric dimension and adjacency dimension.
Theorem 4.1. [4]
Let G be a connected graph of order n . Then(a) dim( G ) = 1 if and only if G = P n ;(b) for n ≥ , dim( G ) = n − if and only if G = K s,t ( s, t ≥ ), G = K s + K t ( s ≥ , t ≥ ), or G = K s + ( K ∪ K t ) ( s, t ≥ );(c) dim( G ) = n − if and only if G = K n . Theorem 4.2. [17]
Let G be a graph of order n . Then(a) adim( G ) = 1 if and only if G ∈ { P , P , P , P , P } ;(b) adim( G ) = n − if and only if G ∈ { K n , K n } . Note that, if f is a resolving broadcast of G with f ( v ) = 2 and f ( w ) = 0 for each w ∈ V ( G ) −{ v } ,then v is an end vertex of P or v is an end vertex of P ∪ P , and adim( P ) = adim( P ∪ P ) = 2 asshown in Theorem 3.9. Also, note that adim( G ) = 2 implies bdim( G ) = 2. So, Observation 2.3(a),Theorems 3.9, 4.1 and 4.2 imply the following proposition. roposition 4.3. Let G be a graph of order n . Then(a) bdim( G ) = 1 if and only if G ∈ { P , P , P , P , P } ;(b) bdim( G ) = 2 if and only if G ∈ H − H as described in Theorem 3.9 (see Figure 2);(c) bdim( G ) = n − if and only if G ∈ { K n , K n } . The next two questions about graphs with high adjacency dimension and broadcast dimensionare both open.
Question 4.4.
What graphs G of order n satisfy adim( G ) = n − ? Question 4.5.
What graphs G of order n satisfy bdim( G ) = n − ? v v u u u u Figure 2:
The graphs G satisfying adim( G ) = 2, where black vertices must be present, a solid edgemust be present whenever the two vertices incident to the solid edge are in the graph, but a dottededge is not necessarily present.A graph is planar if it can be drawn in a plane without edge crossing. For two graphs G and H , H is called a minor of G if H can be obtained from G by vertex deletion, edge deletion, or edgecontraction. Theorem 4.6. [26]
A graph G is planar if and only if neither K nor K , is a minor of G . Remark 4.7.
It was shown in [19] that there exists a non-planar graph G with dim( G ) = 2 .However, adim( G ) = 2 ( bdim( G ) = 2 , respectively) implies G is planar (see Figure 2). Also, notethat, for each k ≥ , there exists a non-planar graph G satisfying bdim( G ) = k and adim( G ) = k ,respectively. For example, the graph G of order n = k + 2 k with bdim( G ) = k ( adim( G ) = k ,respectively) described in the proof of Theorem 3.7 contains K k as a subgraph. Since K k , for k ≥ , contains K as a minor, G is not planar by Theorem 4.6. dim( G ) , adim( G ) , and bdim( G ) Next, we provide a connected graph G such that both adim( G ) − bdim( G ) and bdim( G ) − dim( G )can be arbitrarily large. In fact, we obtain the stronger result that adim( G )bdim( G ) and bdim( G )dim( G ) can bearbitrarily large. We first recall some results on grid graphs. Proposition 5.1. [3]
For the grid graph G = P m × P n ( m, n ≥ ), dim( G ) = 2 . Proposition 5.2. [19]
For the d -dimensional grid graph G = Q di =1 P n i , where d ≥ and n i ≥ foreach i ∈ { , . . . , d } , dim( G ) ≤ d . With Theorem 3.14, propositions 5.1 and 5.2 immediately imply the following corollary.
Corollary 5.3. If G is the grid graph P m × P n ( m, n ≥ ), then bdim( G ) = Θ( m + n ) . Moregenerally, if G is the d -dimensional grid graph Q di =1 P n i with n i ≥ for each i = 1 , . . . , d , then bdim( G ) = Θ( P di =1 n i ) , where the constant in the upper bound depends on d ≥ . heorem 5.4. For k ≥ , let G be the d -dimensional grid graph Q di =1 P k . Then bdim( G ) = Θ( k ) ,and adim( G ) = Θ( k d ) , where the constants in the bounds depend on d . So, adim( G )bdim( G ) and bdim( G )dim( G ) canbe arbitrarily large.Proof. Note that dim( G ) ≤ d by Proposition 5.2 and bdim( G ) = Θ( k ) by Corollary 5.3. To see thatadim( G ) = Θ( k d ), first note that adim( G ) = O ( k d ) since | V ( G ) | = O ( k d ). Moreover, any adjacencyresolving set of G must contain at least one vertex from every Q di =1 P subgraph of G except for atmost one, so adim( G ) = Ω( k d ).In the next result, we show that the multiplicative gap between bdim( G ) and adim( G ) in The-orem 5.4 is tight up to a constant factor. To state this result, we define ∆ ′ ( G ) to be the maximumvalue of t for which there exists a positive integer j and a vertex v ∈ V ( G ) such that there exist atleast t distinct vertices u , . . . , u t ∈ V ( G ) with d G ( u i , v ) = j for each i = 1 , . . . , t . Note that when G = P k × P k , we have ∆ ′ ( G ) = Θ( k ), so adim( G )bdim( G ) = Θ( k ) = Θ(∆ ′ ( G )). Proposition 5.5.
For all graphs G , adim( G )bdim( G ) = O (∆ ′ ( G )) .Proof. Given a resolving broadcast f of G with P v ∈ V ( G ) f ( v ) = bdim( G ), we show how to convert f into an adjacency resolving set for G which uses at most (∆ ′ ( G ) + 1)bdim( G ) vertices. Let v be avertex v ∈ V ( G ) with f ( v ) >
0. If f ( v ) = 1, then we put the vertex v into the adjacency resolving setfor G . If f ( v ) >
1, then for each k >
0, we list the vertices u , . . . , u t with d G ( u i , v ) = k , and we addeach vertex u , . . . , u t into the adjacency resolving set for G , as well as the vertex v . Thus we addat most (∆ ′ ( G ) + 1) vertices to the adjacency resolving set for each vertex v ∈ V ( G ) with f ( v ) > k with k ≤ f ( v ). This implies that adim( G ) ≤ (∆ ′ ( G ) + 1)bdim( G ).The proof of the last proposition also implies the following proposition. Proposition 5.6.
For all graphs G of order n , bdim( G )∆ ′ ( G ) = Ω( n ) . It was shown in [6] that metric dimension is not a monotone parameter on subgraph inclusion;see [6] for an example satisfying H ⊂ G and dim( H ) > dim( G ). Next, we show that for two graphs G and H with H ⊂ G , dim( H ) − dim( G ), bdim( H ) − bdim( G ), and adim( H ) − adim( G ) can bearbitrarily large. In fact, we obtain the stronger result that dim( H )dim( G ) , bdim( H )bdim( G ) , and adim( H )adim( G ) can bearbitrarily large. Theorem 5.7.
There exist connected graphs G and H such that H ⊂ G and dim( H )dim( G ) , bdim( H )bdim( G ) , and adim( H )adim( G ) can be arbitrarily large.Proof. For k ≥
3, let H = K k ( k +1)2 ; let V ( H ) be partitioned into V , V , . . . , V k such that V i = { w i, , w i, , . . . , w i,i } with | V i | = i , where i ∈ { , , . . . , k } . Let G be the graph obtained from H and k isolated vertices u , u , . . . , u k as follows: u is adjacent to V ∪ ( ∪ kj =2 { w j, } ), u is adjacent to V ∪ ( ∪ kj =3 { w j, } ), u is adjacent to V ∪ ( ∪ kj =4 { w j, } ), and so on, i.e., for each i ∈ { , , . . . , k } , u i is adjacent to each vertex of V i ∪ ( ∪ kj = i +1 { w j,i } ) (see the graph G in Figure 3 when k = 4). Sincediam( H ) = 1 and diam( G ) = 2, dim( H ) = bdim( H ) = adim( H ) and dim( G ) = bdim( G ) = adim( G )by Observation 2.3(b). Note that H ⊂ G and dim( H ) = k ( k +1)2 − { u , u , . . . , u k } forms a resolving set of G , dim( G ) ≤ k . So, dim( H )dim( G ) = bdim( H )bdim( G ) = adim( H )adim( G ) ≥ k + k − k → ∞ as k → ∞ .Next we find all trees T for which dim( T ) = bdim( T ). First we recall some terminology. Fixa tree T . An end vertex ℓ is called a terminal vertex of a major vertex v if d ( ℓ, v ) < d ( ℓ, w ) forevery other major vertex w in T . The terminal degree , ter ( v ), of a major vertex v is the number ofterminal vertices of v in T , and an exterior major vertex is a major vertex that has positive terminaldegree. We denote by ex ( T ) the number of exterior major vertices of T , and σ ( T ) the number ofend vertices of T . , w , w , w , w , w , w , w , w , w , u u u u V V V V H = K Figure 3:
A graph G such that H ⊂ G and dim( H )dim( G ) = bdim( H )bdim( G ) = adim( H )adim( G ) can be arbitrarily large;here, k = 4 and H = K for the example described in Theorem 5.7. Theorem 5.8. [4, 19, 21]
For a tree T that is not a path, dim( T ) = σ ( T ) − ex ( T ) . Theorem 5.9. [21]
Let T be a tree with ex ( T ) = k ≥ , and let v , v , . . . , v k be the exterior majorvertices of T . For each i ( ≤ i ≤ k ), let ℓ i, , ℓ i, , . . . , ℓ i,σ i be the terminal vertices of v i with ter ( v i ) = σ i ≥ , and let P i,j be the v i − ℓ i,j path, where ≤ j ≤ σ i . Let W ⊆ V ( T ) . Then W isa minimum resolving set of T if and only if W contains exactly one vertex from each of the paths P i,j − v i ( ≤ j ≤ σ i and ≤ i ≤ k ) with exactly one exception for each i with ≤ i ≤ k and W contains no other vertices of T . Proposition 5.10.
Let T be a non-trivial tree. Then dim( T ) = bdim( T ) if and only if T ∈ { P , P } or T is a tree obtained from the star K ,x ( x ≥ ) by subdividing at most x − edges exactly once.Proof. ( ⇐ ) First, let T ∈ { P , P } , and let ℓ be an end vertex of T . Let g be a function defined on V ( G ) such that g ( ℓ ) = 1 and g ( v ) = 0 for each v ∈ V ( T ) − { ℓ } . Then g is a resolving broadcastof T , and thus bdim( T ) = 1 = dim( T ) by Observation 2.3(a) and Theorem 4.1(a). Second, let T be a tree obtained from the star K ,x ( x ≥
3) by subdividing at most x − w be the major vertex of T , and let ℓ , ℓ , . . . , ℓ x be the terminal vertices of w in T such that d ( w, ℓ ) ≥ d ( w, ℓ ) ≥ . . . ≥ d ( w, ℓ x ); then d ( w, ℓ x ) = 1. If f : V ( T ) → Z + ∪ { } is a function definedby f ( v ) = (cid:26) v ∈ N ( w ) − { ℓ x } f is a resolving broadcast of T , and thus bdim( T ) ≤ x − T ) by Theorem 5.8. ByObservation 2.3(a), bdim( T ) = dim( T ).( ⇒ ) Let dim( T ) = bdim( T ). Let f : V ( T ) → Z + ∪ { } be a resolving broadcast of T with c f ( T ) = dim( T ), and let R = supp T ( f ). First, let ex ( T ) = 0, i.e., T is a path; then c f ( T ) = 1by Theorem 4.1(a). So, b R ( u ) ∈ { , , } for each u ∈ V ( T ). Thus, T ∈ { P , P } . Second, let ex ( T ) = 1. Let v be the exterior major vertex of T with terminal vertices ℓ , ℓ , . . . , ℓ x such that d ( v, ℓ ) ≥ d ( v, ℓ ) ≥ . . . ≥ d ( v, ℓ x ); then x ≥ T ) = x − N ( v ) = ∪ xi =1 { s i } and let s i lie on the v − ℓ i path, where i ∈ { , , . . . , x } . If d ( v, ℓ x ) ≥
2, then thereexists j ∈ { , , . . . , x } such that d ( v, ℓ j ) ≥ b R ( s j ) = b R ( ℓ j ), contradicting the assumption that f is a resolving broadcast of T . So, d ( v, ℓ x ) = 1 and b R ( ℓ x ) = x − . If d ( v, ℓ ) = d ≥
3, say the v − ℓ path is given by v, s , s , . . . , s d = ℓ , then (i) b R ( ℓ ) = b R ( ℓ x ) if f ( s ) = 1; (ii) b R ( s i − ) = b R ( s i +1 )if f ( s i ) = 1 for some i ∈ { , , . . . , d − } ; (iii) b R ( s ) = b R ( ℓ x ) if f ( ℓ ) = 1. So, d ( v, ℓ ) ≤
2. Next,let ex ( T ) ≥
2; we show that bdim( T ) > dim( T ). Let v , v , . . . , v a be distinct exterior major verticesof T , where a ≥
2. Let ℓ , ℓ , . . . , ℓ α be the set of terminal vertices of v , and let ℓ ′ , ℓ ′ , . . . , ℓ ′ β bethe set of terminal vertices of v in T ; let P ,i be the v − ℓ i path excluding v , and let P ,j be the v − ℓ ′ j path excluding v in T . By Theorem 5.9, c f ( P ,x ) = c f ( P ,y ) = 0 for some x ∈ { , , . . . , α } and y ∈ { , . . . , β } . Since b R ( ℓ ,x ) = b R ( ℓ ′ ,y ) = | R | , f fails to be a resolving broadcast of T , andthus bdim( T ) > dim( T ). roposition 5.10 implies the following corollary. Corollary 5.11.
For any non-trivial tree T , dim( T ) = adim( T ) if and only if T ∈ { P , P } or T isa tree obtained from the star K ,x ( x ≥ ) by subdividing at most x − edges exactly once. Throughout this section, let v and e , respectively, denote a vertex and an edge of a connected graph G such that both G − v and G − e are connected graphs. First, we consider the effect of vertexdeletion on adjacency dimension and broadcast dimension. It is known that dim( G ) − dim( G − v )and dim( G − v ) − dim( G ), respectively, can be arbitrarily large; see [1] and [5], respectively. Weshow that bdim( G )bdim( G − v ) can be arbitrarily large, whereas adim( G ) ≤ adim( G − v ) + 1. We also showthat bdim( G − v ) − bdim( G ) and adim( G − v ) − adim( G ) can be arbitrarily large.We recall the following useful result. Proposition 6.1. [9]
Let H be a graph of order n ≥ . Then adim( K + H ) ≥ adim( H ) . Remark 6.2.
The value of bdim( G )bdim( G − v ) can be arbitrarily large, as G varies.Proof. Let G = ( P k × P k ) + K , and let v be the vertex in the K . Then bdim( G − v ) = Θ( k ) byTheorem 5.4, but bdim( G ) = adim( G ) ≥ adim( G − v ) = Θ( k ) by Proposition 6.1 and Theorem 5.4. Proposition 6.3.
For any graph G , adim( G ) ≤ adim( G − v ) + 1 , where the bound is sharp.Proof. Let S be a minimum adjacency resolving set of G − v . Note that, for any vertex x in G − v , a S ( x ) in G − v remains the same in G . So, S ∪ { v } forms an adjacency resolving set of G , and henceadim( G ) ≤ | S | + 1 = adim( G − v ) + 1. For the sharpness of the bound, let G = K n for n ≥
3; thenadim( G ) = n − G − v ) = n −
2, for any v ∈ V ( G ), by Theorem 4.2(b). Remark 6.4.
The value of bdim( G − v ) − bdim( G ) and adim( G − v ) − adim( G ) can be arbitrarilylarge, as G varies.Proof. Let G be the graph in Figure 4, where k ≥
2. Note that diam( G ) = diam( G − v ) = 2; thus,dim( G ) = bdim( G ) = adim( G ) and dim( G − v ) = bdim( G − v ) = adim( G − v ) by Observation 2.3(b).First, we show that dim( G ) = k + 1. Let S be any minimum resolving set of G . Note that, foreach i ∈ { , , . . . , k } , x i and z i are twin vertices of G ; thus | S ∩ { x i , z i }| ≥ S ′ = ∪ ki =1 { x i } ⊆ S . Since r S ′ ( y i ) = r S ′ ( z i ) for each i ∈ { , , . . . , k } , | S | ≥ k + 1; thus dim( G ) ≥ k + 1. On the other hand, S ′ ∪ { v } forms a resolving set of G , and thusdim( G ) ≤ k + 1. So, dim( G ) = k + 1.Second, we show that dim( G − v ) = 2 k . Let R be any minimum resolving set of G − v . Notethat, for each i ∈ { , , . . . , k } , any two vertices in { x i , y i , z i } are twin vertices of G − v . ByObservation 2.1(a), | R ∩ { x i , y i , z i }| ≥ i ∈ { , , . . . , k } ; thus | R | ≥ k . Since ∪ ki =1 { x i , y i } forms a resolving set of G − v , dim( G − v ) ≤ k . Thus, dim( G − v ) = 2 k .Therefore, dim( G − v ) − dim( G ) = bdim( G − v ) − bdim( G ) = adim( G − v ) − adim( G ) =2 k − ( k + 1) = k − → ∞ as k → ∞ .Next, we consider the effect of edge deletion. We recall the following result on the effect of edgedeletion on metric dimension. Theorem 6.5. [5] (a) For any graph G and any edge e ∈ E ( G ) , dim( G − e ) ≤ dim( G ) + 2 . x y z x y z x y z x k y k z k Figure 4:
A graph G such that dim( G − v ) − dim( G ) = bdim( G − v ) − bdim( G ) = adim( G − v ) − adim( G ) can be arbitrarily large, where k ≥ u u u k e Figure 5:
A graph G such that dim( G ) − dim( G − e ) can be arbitrarily large, where k ≥ (b) The value of dim( G ) − dim( G − e ) can be arbitrarily large (see Figure 5). Now, we consider the effect of edge deletion on adjacency dimension. We begin with the followinglemma, which is used in proving Theorem 6.7.
Lemma 6.6.
For any graph G , let e = xy ∈ E ( G ) .(a) If S is an adjacency resolving set of G , then S ∪ { x, y } is an adjacency resolving set of G − e .(b) If R is an adjacency resolving set of G − e , then R ∪ { x, y } is an adjacency resolving set of G .Proof. Let e = xy ∈ E ( G ).(a) Since S is an adjacency resolving set of G , S ′ = S ∪ { x, y } is also an adjacency resolving setof G . Since the adjacency code of each vertex, excluding x and y , with respect to S ′ in G remainsthe same in G − e , S ′ is an adjacency resolving set of G − e .(b) Since R is an adjacency resolving set of G − e , R ′ = R ∪ { x, y } is an adjacency resolving setof G − e . Since the adjacency code of each vertex, excluding x and y , with respect to R ′ in G − e remains the same in G , R ′ is an adjacency resolving set of G . Theorem 6.7.
For any graph G and any edge e ∈ E ( G ) , adim( G ) − ≤ adim( G − e ) ≤ adim( G )+1 . Proof.
We denote by d H, ( x, y ) the adjacency distance between two vertices x and y in a graph H .First, we show that adim( G − e ) ≤ adim( G ) + 1. Let S be a minimum adjacency resolving setof G , and let e ∈ E ( G ). Let x, y ∈ V ( G − e ) − S = V ( G ) − S such that z ∈ S with d G, ( x, z ) = d G, ( y, z ). Without loss of generality, let d G, ( x, z ) = 1 and d G, ( y, z ) = 2; then xz ∈ E ( G ). If d G − e, ( x, z ) = d G − e, ( y, z ), then e = xz . Since z ∈ S , S ∪ { x } forms an adjacency resolving set of G − e by Lemma 6.6(a). Thus adim( G − e ) ≤ | S | + 1 = adim( G ) + 1. econd, we show that adim( G ) − ≤ adim( G − e ). Let R be any minimum adjacency resolvingset of G − e , and let e = uv ∈ E ( G ). If | R ∩ { u, v }| = 0, then each entry of a R ( u ) and a R ( v )is 1 or 2; thus, the adjacency code of each vertex with respect to R in G − e remains the samein G , and hence R is an adjacency resolving set of G . If | R ∩ { u, v }| = 1, say u ∈ R and v R without loss of generality, then R ∪ { v } forms an adjacency resolving set of G by Lemma 6.6(b).If | R ∩ { u, v }| = 2 (i.e., u, v ∈ R ), then R is an adjacency resolving set of G by Lemma 6.6(b).Therefore, adim( G ) ≤ | R | + 1 = adim( G − e ) + 1. Remark 6.8.
The bounds in Theorem 6.7 are sharp.(a) For a graph G satisfying adim( G ) − G − e ) , let G = K n for n ≥ . Then adim( G − e ) = n − and adim( G ) = n − .(b) For a graph G satisfying adim( G − e ) = adim( G ) + 1 , let G be the graph in Figure 6. Let N ( u ) − { u } = ∪ ai =1 { x i } , N ( u ) − { u , u } = ∪ bi =1 { y i } , and N ( u ) − { u } = ∪ ci =1 { z i } , where a, c ≥ and b ≥ .First, we show that adim( G − e ) = a + b + c − . Let S be a minimum adjacency resolving set of G − e . Since any two vertices in ∪ ai =1 { x i } , ∪ bi =1 { y i } , and ∪ ci =1 { z i } , respectively, are twin verticesin G − e , by Observation 2.1(b), we have | S ∩ ( ∪ ai =1 { x i } ) | ≥ a − , | S ∩ ( ∪ bi =1 { y i } ) | ≥ b − and | S ∩ ( ∪ ci =1 { z i } ) | ≥ c − . Let S ′ = ( ∪ ai =2 { x i } ) ∪ ( ∪ bi =2 { y i } ) ∪ ( ∪ ci =2 { z i } ) ⊆ S . Note that (i) a S ′ ( u ) is the ( a + b + c − -vector with 1 on the first ( a − entries and 2 on the rest of the entries; (ii) a S ′ ( u ) is the ( a + b + c − -vector with 1 on the a th through ( a + b − th entries and 2 on the rest ofthe entries; (iii) a S ′ ( u ) is the ( a + b + c − -vector with 2 on the first ( a + b − entries and 1 on therest of the entries; (iv) a S ′ ( x ) = a S ′ ( y ) = a S ′ ( z ) = a + b + c − . Since S ′ fails to be an adjacencyresolving set of G − e and, for any w ∈ V ( G − e ) − S ′ , S ′ ∪ { w } fails to be an adjacency resolvingset of G − e , adim( G − e ) ≥ a + b + c − . On the other hand, S ′ ∪ { x , y } forms an adjacencyresolving set of G − e , and hence adim( G − e ) ≤ a + b + c − . Thus, adim( G − e ) = a + b + c − .Next, let e = x z . We show that adim( G ) = a + b + c − . By Theorem 6.7, adim( G ) ≥ a + b + c − . Since R = ( ∪ ai =1 { x i } ) ∪ ( ∪ bi =2 { y i } ) ∪ ( ∪ ci =2 { z i } ) forms an adjacency resolving set of G with | R | = a + b + c − , adim( G ) ≤ a + b + c − . Thus, adim( G ) = a + b + c − . u u u e Figure 6:
A graph G with adim( G − e ) = adim( G ) + 1. Question 6.9. Is bdim( G − e ) ≤ bdim( G ) + d G − e ( u, v ) − for any graph G , where e = uv ∈ E ( G ) ? Question 6.10.
Is there a family of graphs G such that bdim( G ) − bdim( G − e ) grows arbitrarilylarge? Below are some open problems about broadcast dimension that are only partially answered by theresults in this paper.
Question 7.1.
What graphs G satisfy dim( G ) = bdim( G ) ? Question 7.2.
What graphs G satisfy bdim( G ) = adim( G ) ? Proposition 5.10, Corollary 5.11, and the results in Section 4 make some progress toward an-swering Questions 7.1 and 7.2. uestion 7.3. Is there a family of graphs G k with bdim( G ) = k for which adim( G ) = 2 Ω( k ) ? Theorem 5.4 shows that for each d ≥ G k with bdim( G ) = k forwhich adim( G ) = Ω( k d ). Question 7.4.
What are the values of bdim( T ) and adim( T ) for every tree T ? Proposition 5.10 and Corollary 5.11 make progress on Question 7.4.
Question 7.5.
Is there a polynomial-time algorithm to determine the value of bdim( G ) for everygraph G ? Heggernes and Lokshtanov [14] found a polynomial-time algorithm for computing the broadcastdomination number γ b ( G ). This differs from the standard domination number [11] and some of itsvariants [15, 25], which are NP-hard.Another natural algorithmic problem is to list all minimum resolving broadcasts of a given graph.In the worst-case, any algorithm to solve this problem must take 2 Ω( n ) time for a graph of order n .We find an algorithm that takes 2 O ( n ) time to list all minimum resolving broadcasts of any givengraph of order n . Theorem 7.6.
There is an algorithm that takes O ( n ) time to list all minimum resolving broadcastsof any given graph of order n . Any algorithm for listing all minimum resolving broadcasts of a givengraph of order n must take Ω( n ) time in the worst-case.Proof. For the worst-case, note that the graph H k on 2 k vertices consisting of k copies of K has2 Ω( k ) minimum resolving broadcasts, and so does the graph H ′ k on 2 k + 1 vertices consisting of k copies of K and an isolated vertex, so any algorithm for listing all minimum resolving broadcastsof a given graph of order n must take 2 Ω( n ) on the families H k and H ′ k .For an algorithm to list all minimum resolving broadcasts of any given graph G of order n , welet v , . . . , v n be the vertices of G . Let s = 0 and perform the following steps:1. Increment s . Let S = ∅ .2. For each nonnegative integer solution ( x , . . . , x n ) to the equation x + · · · + x n = s , determineif the function f defined by setting f ( v i ) = x i is a resolving broadcast for G . If f is a resolvingbroadcast, add f to S . If S is nonempty after checking every solution ( x , . . . , x n ), return S and halt. Otherwise go back to step 1.There are (cid:0) s + n − n − (cid:1) nonnegative integer solutions ( x , . . . , x n ) to the equation x + · · · + x n = s ,so the algorithm only checks at most (cid:0) n − n − (cid:1) = 2 O ( n ) solutions for each value of s . For each solution( x , . . . , x n ), it takes polynomial time in n to determine whether the solution corresponds to aresolving broadcast for G . Thus the algorithm has 2 O ( n ) running time for graphs G of order n . Acknowledgements
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