Relations between Metric Dimension and Domination Number of Graphs
aa r X i v : . [ m a t h . C O ] D ec Relations between Metric Dimension andDomination Number of Graphs
Behrooz Bagheri Gh. , Mohsen Jannesari , Behnaz Omoomi
Department of Mathematical SciencesIsfahan University of Technology84156-83111, Isfahan, Iran
Abstract
A set W ⊆ V ( G ) is called a resolving set, if for each two distinct vertices u, v ∈ V ( G ) there exists w ∈ W such that d ( u, w ) = d ( v, w ), where d ( x, y ) is thedistance between the vertices x and y . The minimum cardinality of a resolving setfor G is called the metric dimension of G , and denoted by β ( G ). In this paper, weprove that in a connected graph G of order n , β ( G ) ≤ n − γ ( G ), where γ ( G ) isthe domination number of G , and the equality holds if and only if G is a completegraph or a complete bipartite graph K s,t , s, t ≥
2. Then, we obtain new bounds for β ( G ) in terms of minimum and maximum degree of G . Keywords:
Resolving set; Metric dimension; Dominating set; Domination number.
Throughout the paper, G = ( V, E ) is a finite, simple, and connected graph of order n . Thedistance between two vertices u and v , denoted by d ( u, v ), is the length of a shortest pathbetween u and v in G . The diameter of G , denoted by diam ( G ) is max { d ( u, v ) | u, v ∈ V } .The set of all neighbors of a vertex v is denoted by N ( v ). The maximum degree and minimumdegree of graph G , are denoted by ∆( G ) and δ ( G ), respectively. The notations u ∼ v and u ≁ v denote the adjacency and non-adjacency relations between u and v , respectively.For an ordered set W = { w , w , . . . , w k } ⊆ V ( G ) and a vertex v of G , the k -vector r ( v | W ) := ( d ( v, w ) , d ( v, w ) , . . . , d ( v, w k ))is called the metric representation of v with respect to W . The set W is called a resolving set for G if distinct vertices have different representations. A resolving set for G with minimumcardinality is called a metric basis , and its cardinality is the metric dimension of G , denoted by β ( G ).It is obvious that to see whether a given set W is a resolving set, it is sufficient to consider thevertices in V ( G ) \ W , because w ∈ W is the unique vertex of G for which d ( w, w ) = 0. When W s a resolving set for G , we say that W resolves G . In general, we say an ordered set W resolvesa set T ⊆ V ( G ) of vertices in G , if for each two distinct vertices u, v ∈ T , r ( u | W ) = r ( v | W ).In [12], Slater introduced the idea of a resolving set and used a locating set and the locationnumber for what we call a resolving set and the metric dimension, respectively. He describedthe usefulness of these concepts when working with U.S. Sonar and Coast Guard Loran stations.Independently, Harary and Melter [6] discovered the concept of the location number as welland called it the metric dimension. For more results related to these concepts see [2, 3, 5,8, ? ]. The concept of a resolving set has various applications in diverse areas including coinweighing problems [11], network discovery and verification [1], robot navigation [8], mastermindgame [2], problems of pattern recognition and image processing [10], and combinatorial searchand optimization [11].The following bound is the known upper bound for metric dimension. Theorem A. [4] If G is a connected graph of order n , then β ( G ) ≤ n − diam ( G ) . A set Γ ⊆ V ( G ) is a dominating set for G if every vertex not in Γ has a neighbor in Γ. Adominating set with minimum size is a minimum dominating set for G . The domination number of G , γ ( G ), is the cardinality of a minimum dominating set. In Section 2, we prove that β ( G ) ≤ n − γ ( G ). Moreover, we prove that β ( G ) = n − γ ( G ) if and only if G is a completegraph or a complete bipartite graph K s,t , s, t ≥
2. In Section 3, regarding to known bounds of γ ( G ), we obtain new upper bounds for metric dimension in terms of other graph parameters. In this section, we prove that β ( G ) ≤ n − γ ( G ). Moreover, we show that β ( G ) = n − γ ( G ) ifand only if G is a complete graph or a complete bipartite graph K s,t , s, t ≥ u, v ∈ V ( G ) are called false twin vertices if N ( u ) = N ( v ). Lemma 1.
Let G be a connected graph. Then there exists a minimum dominating set for G which does not have any pair of false twin vertices. Proof.
Let Γ be a minimum dominating set for G with minimum number of false twin pairs ofvertices and u, v be an arbitrary false twin pair in Γ. Since u and v dominate the same verticesin G , they have no neighbors in Γ; otherwise, Γ \ { u } and Γ \ { v } are dominating sets in G which is a contradiction. On the hand, G is connected, hence u and v have some neighbors in V ( G ) \ Γ. Now, Γ ′ = Γ ∪ { x } \ { u } , where x is a neighbor of u in V ( G ) \ Γ, is a dominating setfor G with fewer number of false twin pair of vertices. This contradiction implies that Γ has nofalse twin pair of vertices. Theorem 1.
For every connected graph G of order n , β ( G ) ≤ n − γ ( G ) . In particular, if Γ is aminimum dominating set for G with no false twin pair of vertices, then V ( G ) \ Γ is a resolvingset for G . roof. By Lemma 1, G has a minimum dominating set Γ with no pair of false twin vertices.Suppose, on the contrary, that V ( G ) \ Γ is not a resolving set for G . Then, there exist vertices u and v in Γ such that r ( u | V ( G ) \ Γ) = r ( v | V ( G ) \ Γ). This implies that all neighbors of u and v in V ( G ) \ Γ are the same. Therefore, u and v have no neighbor in Γ; otherwise we can removeone of the vertices u and v from Γ and get a dominating set with cardinality | Γ | −
1. Hence, u and v are false twin vertices, which is a contradiction. Thus, V ( G ) \ Γ is a resolving set for G .Accordingly, β ( G ) ≤ n − γ ( G ) . The following example shows that Theorem 1 gives a better upper bound for β ( G ) comparingthe upper bound in Theorem A. Example 1.
Let G be a connected graph of order k + 1 , k ≥ , obtained from the wheel W k byreplacing each spoke by a path of length three. It is easy to see that γ ( G ) = k + 1 , by Theorem 1, β ( G ) ≤ n − γ ( G ) = 2 k while diam ( G ) ≤ and by Theorem A, β ( G ) ≤ k + 1 − diam ( G ) . In the sequel we need the following definition.
Definition 1.
Let Γ be a dominating set in a connected graph G and u ∈ Γ . A vertex ¯ u ∈ V ( G ) \ Γ is called a private neighbor of u if u is the unique neighbor of ¯ u in Γ , i.e., N (¯ u ) ∩ Γ = { u } . It is clear that each vertex of a minimum dominating set Γ for a graph G has a private neighboror it is a single vertex in Γ. The following lemma provides a minimum dominating set Γ for G with no false twin pair of vertices such that every vertex in Γ has a private neighbor. Lemma 2.
Every connected graph G has a minimum dominating set Γ with no false twin pairof vertices such that every vertex in Γ has a private neighbor. Proof.
By Lemma 1, let Γ be a minimum dominating set with no false twin pair of verticeswith minimum number of single vertices. Also, let u be a single vertex in Γ. Since G is aconnected graph, u has a neighbor in V ( G ) \ Γ, say x . Now Γ ′ = Γ ∪ { x } \ { u } is also a minimumdominating set for G with no false twin pair of vertices, because x is the unique vertex in Γ ′ thatis adjacent to u . Moreover, u is a private neighbor of x in V ( G ) \ Γ ′ . Note that, x was not aprivate neighbor of any vertex in Γ. Therefore, the number of vertices in Γ ′ which have a privateneighbor is more than the number of vertices in Γ which have a private neighbor in V ( G ) \ Γ.On the other words, the number of single vertex in Γ ′ is fewer than Γ. This contradiction impliesthat all vertices in Γ have a private neighbor in V ( G ) \ Γ. Theorem 2.
Let G be a connected graph of order n . Then β ( G ) = n − γ ( G ) if and only if G = K n or G = K s,t , for some s, t ≥ . Proof.
Clearly, for G = K n and G = K s,t , s, t ≥
2, the equality holds. Now let β ( G ) = n − γ ( G ). By Lemma 2, there exists a minimum dominating set Γ for G with no false twinvertices such that all vertices in Γ have a private neighbor in V ( G ) \ Γ. Let Γ = { u , u , . . . , u r } and W = { x , x , . . . , x r } , where x i is private neighbor of u i for an i , 1 ≤ i ≤ r . Since u i is the nique neighbor of x i in Γ, for each i, j , 1 ≤ i, j ≤ r , the i th coordinate of r ( u j | W ) is 1 if andonly if j = i . Therefore, W resolves the set Γ.By Theorem 1, W = V ( G ) \ Γ is a resolving set for G and β ( G ) = n − γ ( G ) implies that W isa metric basis. Now let x ∈ W \ W . Since W resolves Γ, there exists a unique vertex u i ∈ Γ suchthat r ( x | W ) = r ( u i | W ). Thus, x and u i have the same neighbors in W , but N ( u i ) ∩ W = { x i } ,hence N ( x ) ∩ W = { x i } . Thus, W \ W is partitioned into sets V , V , . . . , V r , (some V i ’s couldbe empty) such that for each i , 1 ≤ i ≤ r , and every x ∈ V i , N ( x ) ∩ W = { x i } . Therefore, W is a minimum dominating set for G . Moreover, W has no pair of false twin vertices, becausefor each i , 1 ≤ i ≤ r , x i is the unique neighbor of u i in W . Hence, by Theorem 1 the set B = V ( G ) \ W is a metric basis of G .Now let B i = V i ∪ { u i } . For a fixed i , 1 ≤ i ≤ r , let a be an arbitrary vertex in B i . Since B is a metric basis of G , B \ { a } is not a resolving set for G . Therefore, there exists a vertex x j a ∈ W such that r ( a | B \ { a } ) = r ( x j a | B \ { a } ). If j a = i , then a is adjacent to all vertices in B i \ { a } , since x i is adjacent to all vertices in B i . If j a = i , then a is not adjacent to any vertexin B i , since x j , j = i , is not adjacent to any vertex in B i . Hence, for every two vertices a and a ′ in B i , where j a = j a ′ . Thus, we conclude that, for every vertex a ∈ B i , there exists a vertex x j ∈ W such that r ( a | B \ { a } ) = r ( x j | B \ { a } ), and there are two possibilities j = i or j = i ;in the former case B i is a clique and in the latter case B i is an independent set.Now let there exists i , 1 ≤ i ≤ r , such that for every vertex a ∈ B i , r ( a | B \ { a } ) = r ( x i | B \ { a } ). It was shown that in this case B i is a clique. Moreover, since a is not adjacentto any vertex in W \ { x i } , x i is not adjacent to any vertex in W \ { x i } . Moreover, since x i isnot adjacent to any vertex in B \ B i , a is not adjacent to any vertex in B \ B i . Therefore, theinduced subgraph by B i ∪ { x i } is a maximal connected subgraph of G . Since G is a connectedgraph, G = B i ∪ { x i } , and consequently G = K n .Otherwise, for each i , 1 ≤ i ≤ r , and for every vertex a ∈ B i , r ( a | B \{ a } ) = r ( x i | B \{ a } ) and r ( a | B \ { a } ) = r ( x j | B \ { a } ) for some j = i . Now, for each b ∈ B j , if r ( b | B \ { b } ) = r ( x k | B \ { b } ),then x k is adjacent to all vertices in B i , since b is adjacent to all vertices in B i . Thus, k = i . Itwas shown that in this case each B i , 1 ≤ i ≤ r , is an independent set. Now, since x j is adjacentto all vertices in B j , every vertex a ∈ B i is adjacent to all vertices in B j . Therefore, the inducedsubgraph B i ∪ B j is a complete bipartite graph.Note that, each vertex in B i ∪ B j is not adjacent to any vertex in B k , k ∈ { , , . . . , r } \ { i, j } ,because x i and x j are not adjacent to any vertex in B \ ( B i ∪ B j ). On the other hand, x i and x j are not adjacent to any vertex in W \ { x i , x j } , since all vertices in B i ∪ B j are not adjacentto any vertex in this set. Therefore, the induced subgraph by B i ∪ B j ∪ { x i , x j } is a maximalconnected subgraph of G . Since G is a connected graph, G = B i ∪ B j ∪ { x i , x j } . Furthermore, r ( a | B \ { a } ) = r ( x j | B \ { a } ) implies that x i ∼ x j , because a ∼ x i . Thus, G = K s,t . Since u i ∈ B i and u j ∈ B j , s, t ≥ The domination number is a well studied parameter and there are several bounds for γ ( G )in terms of the other graph parameters. Following the given new upper bound for β ( G ) in heorem 1, several new upper bounds for metric dimension can be obtained. In what follows,we present some of these new upper bounds. Theorem B. [7]
For every graph G of order n and girth g , (i) if g ≥ , then γ ( G ) ≥ δ ( G ) . (ii) if g ≥ , then γ ( G ) ≥ δ ( G ) − . (iii) γ ( G ) ≥ l n G ) m . (iv) If G has degree sequence ( d , d , . . . , d n ) with d i ≥ d i +1 , then γ ( G ) ≥ min { k | k + ( d + d + · · · + d k ) ≥ n } . (v) if δ ( G ) ≥ and g ≥ , then γ ( G ) ≥ ∆( G ) . Theorem C. [9]
Let µ n ≥ µ n − ≥ · · · ≥ µ be the eigenvalues of Laplacian matrix of connectedgraph G of order n ≥ , then γ ( G ) ≥ nµ n ( G ) . By Theorem 1 and above theorems, the list of new upper bounds for metric dimension interms of other graph parameters are obtained.
Corollary 1.
For every connected graph G of order n and girth g , (i) if g ≥ , then β ( G ) ≤ n − δ ( G ) . (ii) if g ≥ , then β ( G ) ≤ n − δ ( G ) + 2 . (iii) β ( G ) ≤ n ( G ) − l n G ) m . (iv) if G has degree sequence ( d , d , . . . , d n ) with d i ≥ d i +1 , then β ( G ) ≤ n − min { k | k + ( d + d + · · · + d k ) ≥ n } . (v) if µ n ≥ µ n − ≥ · · · ≥ µ be the eigenvalues of Laplacian matrix of G , then β ( G ) ≤ n − nµ n ( G ) . (vi) if δ ( G ) ≥ and g ≥ , then β ( G ) ≤ n − ∆( G ) . For each of the given upper bounds in above, infinite classes of graphs can be constructedto show that these bounds could be better than n − diam ( G ).In the following example, we consider the well known graph Kneser KG (2 k + 1 , k ), whichis called odd graph. The Kneser graph with integer parameters n and k , n ≥ k , denoted by KG ( n, k ), is the graph with k element subsets of set { , , . . . , n } as the vertex set and twovertices are adjacent if and only if the corresponding subsets are disjoint. Example 2.
Let G = KG (2 k + 1 , k ) , for k ≥ . Then, n = | V ( G ) | = (cid:0) k +1 k (cid:1) , ∆( G ) = δ ( G ) = k + 1 , g ( G ) = 6 , µ ( k +1 k )( G ) = 2 k + 1 , and diam ( G ) = k . Therefore, we have: (i) β ( G ) ≤ n − k − . ii) β ( G ) ≤ n − k . (iii) β ( G ) ≤ n − (cid:24) ( k +1 k ) k +2 (cid:25) . (iv) β ( G ) ≤ n − ( k +1 k ) k +2 . (v) β ( G ) ≤ n − ( k +1 k ) k +1 . References [1]
Z. Beerliova, F. Eberhard, T. Erlebach, A. Hall, M. Hoffmann, M. Mihal’ak and L.S. Ram ,Network dicovery and verification,
IEEE Journal On Selected Areas in Communications (2006) 2168-2181.[2]
J. Caceres, C. Hernando, M. Mora, I.M. Pelayo, M.L. Puertas, C. Seara and D.R. Wood , On the metric dimension of cartesian products of graphs , SIAM Journal on Discrete Math-ematics (2007) 423-441.[3]
G.G. Chappell, J. Gimbel and C. Hartman , Bounds on the metric and partition dimensionsof a graph , Ars Combinatorics (2008) 349-366.[4] G. Chartrand, L. Eroh, M.A. Johnson and O.R. Ollermann , Resolvability in graphs andthe metric dimension of a graph , Discrete Applied Mathematics (2000) 99-113.[5]
G. Chartrand and P. Zhang , The theory and applications of resolvability in graphs.
A sur-vey. In Proc. 34th Southeastern International Conf. on Combinatorics, Graph Theory andComputing (2003) 47-68.[6]
F. Harary and R.A Melter , On the metric dimension of a graph , Ars Combinatoria (1976)191-195.[7] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater , Fundamentals of domination in graphs,
Marcel Dekker Inc. New York (1998).[8]
S. Khuller, B. Raghavachari and A. Rosenfeld , Landmarks in graphs , Discrete AppliedMathematics (1996) 217-229.[9]
M. Lu, H. Liu, and F. Tian , Bounds of Laplacian spectrum of graphs based on the domi-nation number , Linear Algebra Appl. (2005), 390-396.[10]
R.A. Melter and I. Tomescu , Metric bases in digital geometry , Computer Vision Graphicsand Image Processing (1984) 113-121.[11] A. Sebo and E. Tannier , On metric generators of graphs , Mathematics of Operations Re-search (2004) 383-393.[12]
P.J. Slater , Leaves of trees , Congressus Numerantium (1975) 549-559.(1975) 549-559.