The metric dimension of strong product graphs
Juan A. Rodriguez-Velazquez, Dorota Kuziak, Ismael G. Yero, Jose M. Sigarreta
aa r X i v : . [ m a t h . C O ] M a y The metric dimension of strong product graphs
Juan A. Rodr´ıguez-Vel´azquez (1) , Dorota Kuziak (1) ,Ismael G. Yero (2) and Jos´e M. Sigarreta (3)(1)
Departament d’Enginyeria Inform`atica i Matem`atiques,Universitat Rovira i Virgili, Av. Pa¨ısos Catalans 26, 43007 Tarragona, [email protected], [email protected] (2)
Departamento de Matem´aticas, Escuela Polit´ecnica Superior de AlgecirasUniversidad de C´adiz, Av. Ram´on Puyol s/n, 11202 Algeciras, [email protected] Facultad de Matem´aticas, Universidad Aut´onoma de GuerreroCarlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, M´[email protected]
October 29, 2018
Abstract
For an ordered subset S = { s , s , . . . s k } of vertices and a vertex u in a connectedgraph G , the metric representation of u with respect to S is the ordered k -tuple r ( u | S ) =( d G ( v, s ) , d G ( v, s ) , . . . , d G ( v, s k )), where d G ( x, y ) represents the distance between thevertices x and y . The set S is a metric generator for G if every two different vertices of G have distinct metric representations. A minimum metric generator is called a metricbasis for G and its cardinality, dim ( G ), the metric dimension of G . It is well knownthat the problem of finding the metric dimension of a graph is NP-Hard. In this paperwe obtain closed formulae and tight bounds for the metric dimension of strong productgraphs. Keywords:
Metric generator; metric basis; metric dimension; strong product graph; resolvingset; locating set.
AMS Subject Classification Numbers: A generator of a metric space is a set S of points in the space with the property that everypoint of the space is uniquely determined by its distances from the elements of S . Given asimple and connected graph G = ( V, E ), we consider the metric d G : V × V → N , where1 G ( x, y ) is the length of a shortest path between x and y . ( V, d G ) is clearly a metric space. Avertex v ∈ V is said to distinguish two vertices x and y if d G ( v, x ) = d G ( v, y ). A set S ⊂ V issaid to be a metric generator for G if any pair of vertices of G is distinguished by some elementof S . A metric generator of minimum cardinality is called a metric basis , and its cardinalitythe metric dimension of G , denoted by dim ( G ).The concept of metric dimension was introduced by Slater in [19], where the metric gener-ators were called locating sets , and studied independently by Harary and Melter [8], where themetric generators were called resolving sets . Applications of this invariant to the navigation ofrobots in networks are discussed in [12] and applications to chemistry in [10, 11]. This invariantwas studied further in a number of other papers including for example [2, 3, 4, 6, 17, 20, 21].Several variations of metric generators have been appearing in the literature, like those aboutresolving dominating sets [1], independent resolving sets [5], local metric sets [17], resolvingpartitions [6, 20], and strong metric generators [14, 18].It was shown in [7] that the problem of computing dim ( G ) is NP-hard. This suggestsfinding the metric dimension for special classes of graphs or obtaining good bounds on thisinvariant. Metric basis have been studied, for instance, for digraphs [16], Cartesian productgraphs [2, 20], corona product graphs [14, 21], distance-hereditary graphs [15], and Hamminggraphs [13]. In this paper we study the problem of finding exact values or sharp bounds forthe metric dimension of strong product graphs and express these in terms of invariants of thefactor graphs.The strong product of two graphs G = ( V , E ) and H = ( V , E ) is the graph G ⊠ H =( V, E ), such that V = V × V and two vertices ( a, b ) , ( c, d ) ∈ V are adjacent in G ⊠ H if andonly if • a = c and bd ∈ E or • b = d and ac ∈ E or • ac ∈ E and bd ∈ E .One of our tools will be a well-known result which states the relationship between the distancebetween two vertices in G ⊠ H and the distances between the corresponding vertices in thefactor graphs. Remark 1. [9]
Let G and H be two connected graphs. Then d G ⊠ H (( a, b ) , ( c, d )) = max { d G ( a, c ) , d H ( b, d ) } . We begin with a general upper bound for the metric dimension of strong product graphs.
Theorem 2.
Let G and H be two connected graphs of order n ≥ and n , respectively. Then dim ( G ⊠ H ) ≤ n · dim ( H ) + n · dim ( G ) − dim ( G ) · dim ( H ) . roof. Let V = { u , u , ..., u n } and V = { v , v , ..., v n } be the set of vertices of G and H ,respectively. Let S = ( V × S ) ∪ ( S × V ), where S and S are metric basis for G and H ,respectively. Let ( u i , v j ) and ( u k , v l ) be two different vertices of G ⊠ H . Let u α ∈ S such that u i , u k are distinguished by u α and let v β ∈ S such that v j , v l are distinguished by v β . If i = k ,then ( u i , v j ) and ( u k , v l ) are distinguished by ( u i , v β ) ∈ ( V × S ) ⊂ S . Analogously, if j = l ,then ( u i , v j ) and ( u k , v l ) are distinguished by ( u α , v j ) ∈ ( S × V ) ⊂ S . If i = k and j = l ,then we suppose that neither ( u i , v β ) nor ( u k , v β ) distinguishes the pair ( u i , v j ) , ( u k , v l ), i.e., d G ⊠ H (( u i , v j ) , ( u i , v β )) = d G ⊠ H (( u k , v l ) , ( u i , v β )) (1)and d G ⊠ H (( u i , v j ) , ( u k , v β )) = d G ⊠ H (( u k , v l ) , ( u k , v β )) . (2)By (1) we have d H ( v j , v β ) = max { d G ( u k , u i ) , d H ( v l , v β ) } and since d H ( v j , v β ) = d H ( v l , v β ), weobtain that d H ( v j , v β ) = d G ( u k , u i ) . (3)Also, by (2) we have d H ( v l , v β ) = max { d G ( u i , u k ) , d H ( v j , v β ) } and since d H ( v j , v β ) = d H ( v l , v β ),we obtain that d H ( v l , v β ) = d G ( u i , u k ) . (4)From (3) and (4) we have that d H ( v j , v β ) = d H ( v l , v β ) which is a contradiction with thestatement that v j , v l are distinguished by v β in H .Since K n ⊠ K n ∼ = K n · n and for any complete graph K n , dim ( K n ) = n −
1, we deduce dim ( K n ⊠ K n ) = n · n − n ( n −
1) + n ( n − − ( n − n − n · dim ( K n ) + n · dim ( K n ) − dim ( K n ) · dim ( K n ) . Therefore, the above bound is tight. Examples of non-complete graphs where the above boundis attained can be derived from Theorem 5.Given two vertices x and y in a connected graph G = ( V, E ), the interval I [ x, y ] between x and y is defined as the collection of all vertices which lie on some shortest x − y path. Givena nonnegative integer k we say that G is self k -resolved if for every two different vertices x, y ∈ V , there exists w ∈ V such that • d G ( y, w ) ≥ k and x ∈ I [ y, w ] or • d G ( x, w ) ≥ k and y ∈ I [ x, w ].For instance, the path graphs P n ( n ≥
2) are self (cid:6) n (cid:7) -resolved, the two-dimensional gridgraphs P n (cid:3) P m are self (cid:0) ⌈ n ⌉ + ⌈ m ⌉ (cid:1) -resolved, the hypercube graphs Q k are self k -resolved andthe pseudo sphere graphs S k,r ( k, r ≥
2) are self k -resolved, where S k,r is a graph defined asfollows: we consider r path graphs of order greater than or equal to k + 1 and we identify oneextreme of each one of the r path graphs in one pole a and all the other extreme vertices ofthe paths in a pole b . In particular, S k, is a cycle graph.3 heorem 3. Let H be a self k -resolved graph of order n and let G be a graph of diameter D ( G ) < k . Then dim ( G ⊠ H ) ≤ n · dim ( G ) . Proof.
Let V = { u , u , ..., u n } and V = { v , v , ..., v n } be the set of vertices of G and H ,respectively. Let S be a metric generator for G . We will show that S = S × V is a metricgenerator for G ⊠ H . Let ( u i , v j ) , ( u r , v l ) be two different vertices of G ⊠ H . We differentiatethe following two cases. Case 1. j = l . Since i = r and S is a metric generator for G , there exists u ∈ S suchthat d G ( u i , u ) = d G ( u r , u ). Hence, d G ⊠ H (( u i , v j ) , ( u, v j )) = d G ( u i , u ) = d G ( u r , u ) = d G ⊠ H (( u r , v j ) , ( u, v j )) . Case 2. j = l . Since H is self k -resolved, there exists v ∈ V such that d H ( v, v l ) ≥ k and v j ∈ I [ v, v l ] or d H ( v, v j ) ≥ k and v l ∈ I [ v, v j ]. Say d H ( v, v l ) ≥ k and v j ∈ I [ v, v l ]. In sucha case, for every u ∈ S we have, d G ⊠ H (( u i , v j ) , ( u, v )) = max { d G ( u i , u ) , d H ( v j , v ) } < d H ( v, v l )= max { d G ( u, u r ) , d H ( v, v l ) } = d G ⊠ H (( u r , v l ) , ( u, v )) . Therefore, S is a metric generator for G ⊠ H .Now we derive some consequences of the above result. Corollary 4.
Let n ≥ be an integer. • For any integer n ≥ such that n − < (cid:4) n (cid:5) , dim ( P n ⊠ C n ) ≤ n . • Let k ≥ be an integer. For any self k -resolved graph H of order n , dim ( K n ⊠ H ) ≤ ( n − n . Given a vertex v of a graph G = ( V, E ), we denote by N G ( v ) the open neighborhood of v , i.e., the set of neighbors of v , and by N G [ v ] the closed neighborhood of v , i.e., N G [ v ] = N G ( v ) ∪ { v } . Two vertices u and v are false twins if N G ( u ) = N G ( v ), while they are true twinsif N G [ u ] = N G [ v ]. Note that if two vertices u and v of a graph G = ( V, E ) are (true or false)twins, then d G ( x, u ) = d G ( x, v ), for every x ∈ V − { u, v } . We define the true twin equivalencerelation R on V ( G ) as follows: x R y ↔ N G [ x ] = N G [ y ] .
4f the true twin equivalence classes are U , U , ..., U t , then every metric generator of G mustcontain at least | U i | − U i for each i ∈ { , ..., t } . Therefore, dim ( G ) ≥ t X i =1 ( | U i | −
1) = n − t, where n is the order of G . Theorem 5.
Let G and H be two nontrivial connected graphs of order n and n having t and t true twin equivalent classes, respectively. Then dim ( G ⊠ H ) ≥ n n − t t . Moreover, if dim ( G ) = n − t and dim ( H ) = n − t , then dim ( G ⊠ H ) = n n − t t . Proof.
Let U , U , ..., U t and U ′ , U ′ , ..., U ′ t be the true twin equivalence classes of G and H ,respectively. Since each U i (and U ′ j ) induces a clique and its vertices have identical closedneighborhoods, U i × U ′ j induces a clique in G ⊠ H and its vertices have identical closed neigh-borhoods, i.e. , for every a, c ∈ U i and b, d ∈ U ′ j , N G ⊠ H [( a, b )] = { ( x, y ) : x ∈ N G [ a ] , y ∈ N H [ b ] } = { ( x, y ) : x ∈ N G [ c ] , y ∈ N H [ d ] } = N G ⊠ H [( c, d )] . Hence, V ( G ) × V ( H ) is partitioned as V ( G ) × V ( H ) = t [ j =1 t [ i =1 U i × U ′ j ! , where U i × U ′ j induces a clique in G ⊠ H and its vertices have identical closed neighborhoods.Therefore, the metric dimension of G ⊠ H is at least P t j =1 (cid:0)P t i =1 ( | U i || U j | − (cid:1) = n n − t t . Finally, if dim ( G ) = n − t and dim ( H ) = n − t , then the above bound and Theorem2 lead to dim ( G ⊠ H ) = n n − t t . As an example of non-complete graph G of order n having t true twin equivalent classes,where dim ( G ) = n − t , we take G = K + ( l [ i =1 K r i ), r i ≥ l ≥
2. In this case G has t = l + 1true twin equivalent classes, n = 1 + P li =1 r i and dim ( G ) = P li =1 ( r i −
1) = n − t . Corollary 6.
Let H be a graph of order n . Let G be a nontrivial connected graph of order n having t true twin equivalent classes. Then dim ( G ⊠ H ) ≥ n ( n − t ) . Theorem 7.
Let H be a self k -resolved graph of order n and let G be a nontrivial connectedgraph of order n having t true twin equivalent classes and diameter D ( G ) < k . If dim ( G ) = n − t , then dim ( G ⊠ H ) = n ( n − t ) . Lemma 8.
A nontrivial connected graph is self -resolved if and only if it does not have truetwin vertices.Proof. Necessity. Let G be a 2-resolved graph. Let x and y be two adjacent vertices in G .Without loss of generality, we take w ∈ V ( G ) such that 2 ≤ k = d G ( x, w ) and y ∈ I [ x, w ]. So,there exists a shortest path x, y, u , ..., u k − , w from x to w and, as a consequence, u ∈ N G [ y ]and u N G [ x ]. Therefore, G does not have true twin vertices.Sufficiency. If for every u, v ∈ V ( G ), N G [ u ] = N G [ v ], then for each pair of adjacent vertices x and y , there exists w ∈ V ( G ) − { x, y } such that d G ( x, w ) = 2 and y ∈ I [ x, w ] or d G ( y, w ) = 2and x ∈ I [ y, w ]. On the other hand, if d G ( u, v ) ≥
2, then for w = u we have d G ( v, w ) ≥ u ∈ I [ v, w ]. Therefore, G is self 2-resolved.By Lemma 8 we deduce the following consequence of Theorem 7. Corollary 9.
Let H be a connected graph of order n ≥ . If H does not have true twinvertices and n ≥ , then dim ( K n ⊠ H ) = n ( n − . The following remark emphasizes some particular cases of the above result.
Remark 10.
Let n ≥ be an integer. • For any tree T of order n ≥ , dim ( K n ⊠ T ) = n ( n − . • For any n ≥ , dim ( K n ⊠ C n ) = n ( n − . • For any hypercube Q r = K (cid:3) · · · (cid:3) K | {z } r , r ≥ , dim ( K n ⊠ Q r ) = 2 r ( n − . • For any integers m, n ≥ , dim ( K n ⊠ ( P n (cid:3) P m )) = n · m · ( n − . Now we proceed to study the strong product of path graphs.
Theorem 11.
For any integers n and n such that ≤ n < n , (cid:22) n + n − n − (cid:23) ≤ dim ( P n ⊠ P n ) ≤ (cid:24) n + n − n − (cid:25) . Proof.
Let V = { u , u , ..., u n } and V = { v , v , ..., v n } be the set of vertices of P n and P n , respectively. With the above notation we suppose that two consecutive vertices of V i areadjacent, i ∈ { , } . 6et α = l n − n − m −
1. We define the set S of cardinality l n + n − n − m as follows: S = { ( u , v ) , ( u n , v n ) , ( u , v n − ) , ( u n , v n − ) , ..., ( u , v α ( n − ) , ( u n , v n ) } if l n − n − m is odd, and S = { ( u , v ) , ( u n , v n ) , ( u , v n − ) , ( u n , v n − ) , ..., ( u n , v α ( n − ) , ( u , v n ) } if l n − n − m is even. We will show that S is a metric generator for P n ⊠ P n . Let ( u i , v j ) , ( u k , v l )be two different vertices of P n ⊠ P n . We differentiate two cases.Case 1. j = l . We suppose, without loss of generality, that i < k . If j ∈ { , ..., n } and d P n ⊠ P n (( u i , v j ) , ( u n , v n )) = d P n ⊠ P n (( u k , v j ) , ( u n , v n )), then from max { n − i, n − j } =max { n − k, n − j } we have n − j ≥ n − i > n − k . Hence, j < k and, as a consequence, d P n ⊠ P n (( u i , v j ) , ( u , v )) = max { i − , j − } < k −
1= max { k − , j − } = d P n ⊠ P n (( u k , v j ) , ( u , v )) . Thus, if j ∈ { , ..., n } , then we deduce r (( u i , u j ) | S ) = r (( u k , u j ) | S ).An analogous procedure can be used to show that for j ∈ { t ( n −
1) + 1 , ..., ( t + 1)( n −
1) + 1 } , where t ∈ { , .., α − } , and for j ∈ { α ( n − , ..., n } , it follows r (( u i , u j ) | S ) = r (( u k , u j ) | S ).Case 2. j = l . We suppose, without loss of generality, that j < l and we differentiate twosubcases.Subcase 2.1. l < n . Since ( u , v ) , ( u n , v n ) ∈ S , we only must consider the case when d P n ⊠ P n (( u i , v j ) , ( u , v )) = d P n ⊠ P n (( u k , v l ) , ( u , v ))and d P n ⊠ P n (( u i , v j ) , ( u n , v n )) = d P n ⊠ P n (( u k , v l ) , ( u n , v n )) . In such a situation, since j < l , we have k < i . Hence, d P n ⊠ P n (( u i , v j ) , ( u , v n ) = max { i − , n − j } > max { k − , n − l } = d P n ⊠ P n (( u k , v l ) , ( u , v n ) . So, if ( u , v n ) ∈ S , then r (( u i , v j ) | S ) = r (( u k , v l ) | S ). Moreover, if ( u , v n ) S , then( u , v n − ) ∈ S . Hence, from d P n ⊠ P n (( u i , v j ) , ( u , v n − ) = max { i − , n − − j } = 2 n − − j> n − − l = max { k − , n − − l } = d P n ⊠ P n (( u k , v l ) , ( u , v n − ) ,
7e have r (( u i , v j ) | S ) = r (( u k , v l ) | S ).Subcase 2.2. l ≥ n . We have, d P n ⊠ P n (( u k , v l ) , ( u , v )) = max { k − , l − } = l − > max { i − , j − } = d P n ⊠ P n (( u i , v j ) , ( u , v )) . Thus, in this case r (( u i , v j ) | S ) = r (( u k , v l ) | S ) as well.We conclude that S is a metric generator for P n ⊠ P n and, as a consequence, the upperbound follows.We will show that dim ( P n ⊠ P n ) ≥ j n + n − n − k by contradiction. Let n − x ( n − y ,where n − > y ≥
0. Now we suppose that there exists a metric generator for P n ⊠ P n , say S ′ , of cardinality x . Note that a vertex ( u r , v t ) ∈ S ′ distinguishes two vertices ( u , v j ) , ( u , v j )if and only if | t − j | < r −
1. Analogously, a vertex ( u r , v t ) ∈ S ′ distinguishes two vertices( u n − , v j ′ ) , ( u n , v j ′ ) if and only if | t − j ′ | < n − r . Hence, a vertex of ( u r , v t ) ∈ S ′ distinguishes,at most, 2 n − u , v j ) , ( u , v j ) or ( u n − , v j ′ ) , ( u n , v j ′ ). Thus, if S ′ is a metric generator, then 2 n − x ≤ (2 n − x and, as a consequence, n − ≤ x ( n − − Conjecture 12.
For any integers n and n such that ≤ n < n , dim ( P n ⊠ P n ) = (cid:24) n + n − n − (cid:25) . Theorem 13.
For any integer n ≥ , dim ( P n ⊠ P n ) = 3 .Proof. Let V = { v , v , ..., v n } be the set of vertices of P n . Now, with the above notation, wesuppose that two consecutive vertices of V are adjacent. We will show that S ′ = { ( u , v ) , ( u n , v ) , ( u n , v n ) } is a metric generator for P n ⊠ P n . Let ( u i , v j ) , ( u k , v l ) be two different vertices of P n ⊠ P n .We only must consider the case when d P n ⊠ P n (( u i , v j ) , ( u , v )) = d P n ⊠ P n (( u k , v l ) , ( u , v )) and d P n ⊠ P n (( u i , v j ) , ( u n , v n )) = d P n ⊠ P n (( u k , v l ) , ( u n , v n )). In such a case, if j < l , then k < i and,as a consequence, d P n ⊠ P n (( u i , v j ) , ( u n , v ) = max { n − i, j − } < max { n − k, l − } = d P n ⊠ P n (( u k , v l ) , ( u n , v )) . Analogously, if j > l , then we have d P n ⊠ P n (( u i , v j ) , ( u n , v ) > d P n ⊠ P n (( u k , v l ) , ( u n , v )) . Weconclude that S ′ is a metric generator for P n ⊠ P n and, as a consequence, dim ( P n ⊠ P n ) ≤ dim ( P n ⊠ P n ) ≥
3, we suppose that there exists a metric generator for P n ⊠ P n of cardinality two. Since (0 ,
0) is not a possible distance vector, and the diameter of P n ⊠ P n is n −
1, there are n − P n ⊠ P n is n , acontradiction. So, dim ( P n ⊠ P n ) ≥ dim ( P n ⊠ P n ) = 3.8he following claim will be useful in the proof of Theorem 15. Claim 14.
Let C be a cycle graph. If x, y, u and v are vertices of C such that x and y areadjacent and d C ( u, x ) = d C ( v, x ) , then d C ( u, y ) = d C ( v, y ) . Theorem 15.
For any integers n and n such that n − ≥ j n k ≥ , dim ( P n ⊠ C n ) ≤ n . Proof.
Let V = { u , u , ..., u n − } and V = { v , v , ..., v n − } be the set of vertices of P n and C n , respectively. Here we suppose that v and v n − are adjacent vertices in C n and, withthe above notation, two consecutive vertices of V i are adjacent, i ∈ { , } . Let S be the setof vertices of P n ⊠ C n of the form ( u i , v i ), where the subscript of the second component istaken modulo n . We will show that S is a metric generator for P n ⊠ C n . To begin with, weconsider two different vertices ( u i , v j ) and ( u k , v l ) of P n ⊠ C n . First we consider the case i = k and we suppose, without loss of generality, that j < l . Now, if d P n ⊠ C n (( u i , v j ) , ( u i , v i )) = d P n ⊠ C n (( u i , v l ) , ( u i , v i )), then d C n ( v j , v i ) = d C n ( v l , v i ). So, for i = 0, Claim 14 leads to d P n ⊠ C n (( u , v j ) , ( u , v )) = max { , d C n ( v j , v ) }6 = max { , d C n ( v l , v ) } = d P n ⊠ C n (( u , v l ) , ( u , v )) . Analogously, for i = 0, Claim 14 leads to d P n ⊠ C n (( u i , v j ) , ( u i − , v i − )) = max { , d C n ( v j , v i − ) }6 = max { , d C n ( v l , v i − ) } = d P n ⊠ C n (( u i , v l ) , ( u i − , v i − )) . Hence, r (( u i , v j ) | S ) = r (( u i , v l ) | S ). Now we consider the case i = k . We suppose, without lossof generality, that i < k . If k ≤ (cid:4) n (cid:5) , then n − − i > (cid:4) n (cid:5) = D ( C n ). Thus, d P n ⊠ C n (( u i , v j ) , ( u n − , v n − )) = max { d P n ( u i , u n − ) , d C n ( v j , v n − ) } = max { n − − i, d C n ( v j , v n − } > max { n − − k, d C n ( v l , v n − ) } = d P n ⊠ C n (( u k , v l ) , ( u n − , v n − )) . Moreover, if k > (cid:4) n (cid:5) , then d P n ⊠ C n (( u i , v j ) , ( u , v )) = max { d P n ( u i , u ) , d C n ( v j , v ) } = max { i, d C n ( v j , v ) } < max { k, d C n ( v l , v ) } = d P n ⊠ C n (( u k , v l ) , ( u , v )) . Hence, r (( u i , v j ) | S ) = r (( u k , v l ) | S ). Therefore, the set S of cardinality n is a metric generatorfor P n ⊠ C n . 9 eferences [1] R. C. Brigham, G. Chartrand, R. D. Dutton, and P. Zhang, Resolving domination ingraphs, Mathematica Bohemica (1) (2003) 25–36.[2] J. C´aceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, C. Seara, and D. R. Wood,On the metric dimension of Cartesian product of graphs,
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