On the packing numbers in graphs
aa r X i v : . [ m a t h . C O ] J u l On the packing numbers in graphs
Doost Ali Mojdeh and Babak Samadi ∗ Department of MathematicsUniversity of Mazandaran, Babolsar, Iran [email protected]@gmail.com
Abdollah KhodkarDepartment of MathematicsUniversity of West GeorgiaCarrollton, GA 30118 USA [email protected]
Hamid Reza GolmohammadiDepartment of MathematicsUniversity of Tafresh, Tafresh, IRI [email protected]
Abstract
In this paper, we find upper bounds on the open packing and k -limited packing numbers with emphasis on the cases k = 1 and k = 2. We solve the problem of characterizing all connected graphson n vertices with ρ o ( G ) = n/δ ( G ) which was presented in 2015 byHamid and Saravanakumar. Also, by establishing a relation betweenthe k -limited packing number and double domination number weimprove two upper bounds given by Chellali and Haynes in 2005. Keywords: domination in graphs, tuple dominating sets in graphs,limited packing sets in graphs
AMS Subject Classifications: ∗ Corresponding author Introduction
Throughout this paper, let G be a finite graph with vertex set V = V ( G ),edge set E = E ( G ), minimum degree δ = δ ( G ) and maximum degree∆ = ∆( G ). We use [10] for terminology and notation which are not definedhere. For any vertex v ∈ V ( G ), N ( v ) = { u ∈ G | uv ∈ E ( G ) } denotes the open neighbourhood of v of G , and N [ v ] = N ( v ) ∪ { v } denotes its closedneighbourhood .A subset B ⊆ V ( G ) is a packing (an open packing ) in G if for every distinctvertices u, v ∈ B , N [ u ] ∩ N [ v ] = ∅ ( N ( u ) ∩ N ( v ) = ∅ ). The packing number ρ ( G ) ( open packing number ρ o ( G )) is the maximum cardinality of a packing(an open packing) in G . These concepts have been studied in [7, 8], andelsewhere.In [5], Harary and Haynes introduced the concept of tuple dominationnumbers. Let 1 ≤ k ≤ δ ( G ) + 1. A set D ⊆ V ( G ) is a k -tuple dominatingset in G if | N [ v ] ∩ D | ≥ k , for all v ∈ V ( G ). The k -tuple domination number ,denoted γ × k ( G ), is the smallest number of vertices in a k -tuple dominatingset. In fact, the authors showed that every graph G with δ ≥ k − k -tuple dominating set and hence a k -tuple domination number. When k = 2, γ × ( G ) is called double domination number of G . For the specialcase k = 1, γ × ( G ) = γ ( G ) is the well known domination number (see[6]). The concept of tuple domination has been studied by several authorsincluding [3, 9]. In general, the reader can find a comprehensive informationon various domination parameters in [1] and [6].Gallant et al. [3] introduced the concept of k -limited packing in graphsand exhibited some real-world applications of it to network security, marketsaturation and codes. A set of vertices B ⊆ V is called a k -limited packingset in G if | N [ v ] ∩ B | ≤ k for all v ∈ V , where k ≥
1. The k -limited packingnumber , L k ( G ), is the largest number of vertices in a k -limited packing set.When k = 1 we have L ( G ) = ρ ( G ).In this paper, we find upper bounds on the k -limited packing numbers. InSection 2, we prove that 2( n − ℓ + sδ ∗ ) / (1 + δ ∗ ) is a sharp upper bound on L ( G ) for a connected graph G on n ≥ ℓ , s and δ ∗ = δ ∗ ( G )are the number of pendant vertices, the number of support vertices andmin { deg( v ) | v is not a pendant vertex } , respectively. Also, we give anupper bound on L k ( G ) (with characterization of all graphs attaining it)in terms of the order, size and k . In Section 3, we exhibit a solution tothe problem of characterizing all connected graphs of order n ≥ ρ o ( G ) = n/δ ( G ) posed in [4]. Moreover, we prove that γ × ( G ) + ρ ( G ) ≤ n − δ ( G ) + 2 when δ ( G ) ≥
2. This improves two results in [2] given byChellali and Haynes, simultaneously.2
Main results
The 2-limited packing number of G has been bounded from above by2 n/ ( δ ( G ) + 1) (see [9], as the special case k = 2). We present the followingupper bound which works better for all graphs with pendant vertices, espe-cially trees. First, we recall that a support vertex is called a weak supportvertex if it is adjacent to just one pendant vertex. Theorem 2.1.
Let G be a connected graph of order n ≥ with s supportvertices and ℓ pendant vertices. Then, L ( G ) ≤ n − ℓ + sδ ∗ ( G ))1 + δ ∗ ( G ) and this bound is sharp. Here δ ∗ ( G ) is the minimum degree taken over allvertices which are not pendant vertices.Proof. Let { u , . . . , u s } be the set of weak support vertices in G . Let G ′ be the graph of order n ′ formed from G by adding new vertices v , . . . , v s and edges u v , . . . , u s v s to G (we note that G = G ′ if G has no weaksupport vertex). Clearly s ′ = s, n ′ = n + s and ℓ ′ = ℓ + s (1)in which s ′ and ℓ ′ are the number of support vertives and pendant verticesof G ′ , respectively. Moreover, since n ≥ G is a connected graph, G and G ′ have the same set of vertices of degree at least two. Therefore, δ ∗ ( G ′ ) = δ ∗ ( G ) = δ ∗ . (2)Let B ′ be a maximum 2-limited packing in G ′ . Suppose to the contrarythat there exists a support vertex u in G ′ for which | N [ u ] ∩ B ′ | ≤
1. Thus,there exists a pendant vertex v / ∈ B ′ adjacent to u . It is easy to see that B ′ ∪ { v } is a 2-limited packing in G ′ which contradicts the maximality of B ′ . So, we may always assume that B ′ contains two pendant vertices ateach support vertex. This implies that all support vertices and the other ℓ u − u belong to V ( G ′ ) \ B ′ , inwhich ℓ u is the number of pendant vertices adjacent to u . Moreover, thesependant vertices have no neighbors in B ′ . Therefore, | [ B ′ , V ( G ′ ) \ B ′ ] | ≤ n ′ − | B ′ | − ℓ ′ + 2 s ′ ) . (3)On the other hand, each pendant vertex in B ′ has exactly one neighbor in V ( G ′ ) \ B ′ and each of the other vertices in V ( G ′ ) \ B ′ has at least δ ∗ ( G ′ ) − B ′ . Therefore,( | B ′ | − s ′ )( δ ∗ ( G ′ ) −
1) + 2 s ′ ≤ | [ B ′ , V ( G ′ ) \ B ′ ] | . (4)3ogether inequalities (3) and (4) imply that | B ′ | ≤ n ′ − ℓ ′ + s ′ δ ∗ ( G ′ ))1 + δ ∗ ( G ′ ) . (5)We now let B be a maximum 2-limited packing in G . Clearly, B is a 2-limited packing in G ′ , as well. Thus, | B | ≤ | B ′ | . By (1),(2) and (5) wehave L ( G ) = | B | ≤ | B ′ | ≤ n − ℓ + sδ ∗ )1 + δ ∗ , as desired.To show that the upper bound is sharp, we consider the star K ,n − , for n ≥
3, with L ( K ,n − ) = 2.It is easy to see that L k ( G ) = n if and only if k ≥ ∆( G ) + 1. So, in whatfollows we may always assume that k ≤ ∆( G ) when we deal with L k ( G ).The following theorem provides an upper bound on L k ( G ) of a graph G interms of its order, size and k . Also, we bound ρ o ( G ) from above just interms of the order and size.First, we define Ω and Σ to be the families of all graphs G having thefollowing properties, respectively.( p ) There exists a clique S such that G [ V ( G ) \ S ] is ( k − S has exactly k neighbors in V ( G ) \ S .( p ) There exists a clique S such that G [ V ( G ) \ S ] is a disjoint union ofcopies of K and every vertex in S has exactly one neighbor in V ( G ) \ S . Theorem 2.2.
Let G be a graph of order n and size m . If k ≤ n −√ n − n − m ) or δ ( G ) ≥ k − , then L k ( G ) ≤ n + k/ − p k / − k ) n + 2 m with equality if and only if G ∈ Ω .Furthermore, ρ o ( G ) ≤ n −√ m − n for any graph G with no isolated vertex.The bound holds with equality if and only if G ∈ Σ .Proof. Let L be a maximum k -limited packing set in G and let E ( G [ L ])and E ( G [ V \ L ]) be the edge set of subgraphs of G induced by L and V \ L ,respectively. Clearly, m = | E ( G [ L ]) | + | [ L, V ( G ) \ L ] | + | E ( G [ V \ L ]) | . (6)Therefore,2 m ≤ ( k − | L | + 2 k ( n − | L | ) + ( n − | L | )( n − | L | − . (7)4olving the above inequality for | L | we obtain L k ( G ) = | L | ≤ n + k − p k + 4(1 − k ) n + 8 m , as desired (note that k ≤ n − √ n − n − m ) or δ ( G ) ≥ k − k / − k ) n + 2 m ≥ | E ( G [ L ]) | = ( k − | L | , | [ L, V ( G ) \ L ] | = k ( n − | L | ) and | E ( G [ V ( G ) \ L ]) | =( n − | L | )( n − | L | − V ( G ) \ L is a clique satisfyingthe property ( p ). Thus, G ∈ Ω. Conversely, suppose that G ∈ Ω. Let S be a clique of the minimum size among all cliques having the property( p ). Then, it is easy to see that L = V ( G ) \ S is a k -limited packing forwhich the upper bound holds with equality.The proof of the second result is similar to the proof of the first one when k = 1. k = 1 Hamid and Saravanakumar [4] proved that ρ o ( G ) ≤ nδ ( G ) (8)for any connected graph G of order n ≥
2. Moreover, the authors charac-terized all the regular graphs which attain the above bound. In general,they posed the problem of characterizing all connected graphs of order n ≥ G constructedas follows. Let H be disjoint union of t ≥ K . Join every ver-tex u of H to k new vertices as its private neighbors lying outside V ( H ).Let V = V ( H ) ∪ ( ∪ u ∈ V ( H ) pn ( u )), in which pn ( u ) is the set of neighbors(private neighbors) of u which lies outside V ( H ). Add new edges amongthe vertices in ∪ u ∈ V ( H ) pn ( u ) to construct a connected graph G on the setof vertices in V = V ( G ) with deg( v ) ≥ k + 1, for all v ∈ ∪ u ∈ V ( H ) pn ( u ).Clearly, every vertex in V ( H ) has the minimum degree δ ( G ) = k + 1 andevery vertex in ∪ u ∈ V ( H ) pn ( u ) has exactly one neighbor in V ( H ).We are now in a position to present the following theorem. Theorem 3.1.
Let G be a connected graph of order n ≥ . Then, ρ o ( G ) = nδ ( G ) if and only if G ∈ Γ .Proof. We first state a proof for (8). Let B be a maximum open packing in G . Every vertex in V ( G ) has at most one neighbor in B and hence every5ertex in B has at least δ ( G ) − V ( G ) \ B , by the definitionof an open packing. Thus,( δ ( G ) − | B | ≤ | [ B, V ( G ) \ B ] | ≤ n − | B | . (9)Therefore, ρ o ( G ) = | B | ≤ nδ ( G ) .Considering (9), we can see that the equality in (8) holds if and only ( δ ( G ) − | B | = | [ B, V ( G ) \ B ] | and | [ B, V ( G ) \ B ] | = n − | B | . Since, B is an openpacking, this is equvalent to the fact that H = G [ B ] is a disjoint unionof t = | B | / K , in which every vertex has the minimum degreeand is adjacent to k = δ ( G ) − V ( G ) \ B and each vertex in V ( G ) \ B has exactly one neighbor in B . Now, it is easy to see that theequality in (8) holds if and only G ∈ Γ. Remark 3.2.
Similar to the proof of Theorem 3.1 we have ρ ( G ) ≤ n/ ( δ ( G )+1), for each connected graph G of order n . Furthermore, the characteriza-tion of graphs G attaining this bound can be obtained in a similar fashionby making some changes in Γ. It is sufficient to consider H as a subgraph of G with no edges in which every vertex has exactly δ ( G ) private neighborslying outside V ( H ).In [2], Chellali and Haynes proved that for any graph G of order n with δ ( G ) ≥ γ × ( G ) + ρ ( G ) ≤ n. Also, they proved that γ × ( G ) ≤ n − δ ( G ) + 1for any graph G with no isolated vertices.We note that the second upper bound is trivial for δ ( G ) = 1. So, wemay assume that δ ( G ) ≥
2. In the following theorem, using the conceptsof double domination and k -limited packing, we improve these two upperbounds, simultaneously. Theorem 3.3.
Let G be a graph of order n . If δ ( G ) ≥ , then γ × ( G ) + ρ ( G ) ≤ n − δ ( G ) + 2 . Furthermore, this bound is sharp.Proof.
Let B be a maximum ( δ ( G ) − G . Everyvertex in B has at most δ ( G ) − B . Therefore it has at leasttwo neighbours in V ( G ) \ B . On the other hand, every vertex in V ( G ) \ B has at most δ ( G ) − B , hence it has at least one neighbour6n V ( G ) \ B . This implies that V ( G ) \ B is a double dominating set in G .Therefore, γ × ( G ) + L δ ( G ) − ( G ) ≤ n. (10)Now let 1 ≤ k ≤ ∆( G ) and let B be a maximum k -limited packing set in G . Then | N [ v ] ∩ B | ≤ k , for all v ∈ V ( G ). We claim that B = V ( G ).If B = V ( G ) and u ∈ V ( G ) such that deg( u ) = ∆( G ), then ∆( G ) + 1 = | N [ u ] ∩ B | ≤ k ≤ ∆( G ), a contradiction. Now let u ∈ V ( G ) \ B . It iseasy to check that | N [ v ] ∩ ( B ∪ { u } ) | ≤ k + 1, for all v ∈ V ( G ). Therefore B ∪ { u } is a ( k + 1)-limited packing set in G . Hence L k +1 ( G ) ≥ | B ∪ { u }| = | B | + 1 = L k ( G ) + 1 , for k = 1 , . . . , ∆( G ). Applying this inequality repeatedly leads to L δ − ( G ) ≥ L ( G ) + δ ( G ) − ρ ( G ) + δ ( G ) − . Hence, γ × ( G ) + ρ ( G ) ≤ n − δ ( G ) + 2 by (10). Finally, the upper bound issharp for the complete graph K n with n ≥ References [1] M. Chellali, O. Favaron, A. Hansberg and L. Volkmann, k -Dominationand k -Independence in Graphs: A Survey , Graphs Combin. (2012),1-55.[2] M. Chellali and T.W. Haynes, On paired and double domination ingraphs , Utilitas Math. (2005), 161–171.[3] R. Gallant, G. Gunther, B.L. Hartnell and D.F. Rall, Limited packingin graphs , Discrete Appl. Math. (2010), 1357–1364.[4] S. Hamid and S. Saravanakumar,