A characterization of some graphs with metric dimension two
aa r X i v : . [ m a t h . C O ] S e p A characterization of some graphs with metric dimensiontwo
Ali Behtoei a ∗ , Akbar Davoodi b † , Mohsen Jannesari c ‡ and Behnaz Omoomi d § a Department of Mathematics, Imam Khomeini International University, 34149-16818, Qazvin, Iran b,d
Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran c University of Shahreza, 86149-56841, Shahreza, Iran
Abstract
A set W ⊆ V ( G ) is called a resolving set, if for each pair of distinct vertices u, v ∈ V ( G ) there exists t ∈ W such that d ( u, t ) = d ( v, t ), where d ( x, y ) is the distancebetween vertices x and y . The cardinality of a minimum resolving set for G is calledthe metric dimension of G and is denoted by dim M ( G ). A k -tree is a chordal graphall of whose maximal cliques are the same size k + 1 and all of whose minimal cliqueseparators are also all the same size k . A k -path is a k -tree with maximum degree2 k , where for each integer j , k ≤ j < k , there exists a unique pair of vertices, u and v , such that deg( u ) = deg( v ) = j . In this paper, we prove that if G is a k -path,then dim M ( G ) = k . Moreover, we provide a characterization of all 2-trees with metricdimension two. Throughout this paper all graphs are finite, simple and undirected. The notions δ , ∆ and N G ( v ) stand for minimum degree, maximum degree and the set of neighbours of vertex v in G , respectively.For an ordered set W = { w , w , . . . , w k } of vertices and a vertex v in a connected graph 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 , where d ( x, y ) is the distance between two vertices x and y . Theset W is called a resolving set for G if distinct vertices of G have distinct representationswith respect to W . We say a set S ⊆ V ( G ) resolves a set T ⊆ V ( G ) if for each pairof distinct vertices u and v in T there is a vertex s ∈ S such that d ( u, s ) = d ( v, s ). Aminimum resolving set is called a basis and the metric dimension of G , dim M ( G ), is thecardinality of a basis for G . A graph with metric dimension k is called k - dimensional . ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] N P -complete problem, but the metricdimension of trees can be obtained by a polynomial time algorithm.It is obvious that for every graph G of order n , 1 ≤ dim M ( G ) ≤ n −
1. Chartrand etal. [5] proved that for n ≥
2, dim M ( G ) = n − G is the complete graph K n . They also provided a characterization of graphs of order n and metric dimension n − n − M ( G ) = 1 if and only if G is a path. Moreover,in [12] some properties of 2-dimensional graphs are obtained. Theorem 1.1 [12]
Let G be a -dimensional graph. If { a, b } is a basis for G , then1. there is a unique shortest path P between a and b ,2. the degrees of a and b are at most three,3. the degree of each internal vertex on P is at most five. A chordal graph is a graph with no induced cycle of length greater than three. A k -tree is a chordal graph that all of whose maximal cliques are the same size k + 1 and all ofwhose minimal clique separators are also all the same size k . In other words, a k -tree maybe formed by starting with a set of k + 1 pairwise adjacent vertices and then repeatedlyadding vertices in such a way that each added vertex has exactly k neighbours that forma k -clique.By the above definition, it is clear that if G is a k -tree, then δ ( G ) = k . 1-trees are the sameas trees; 2-trees are maximal series-parallel graphs [4] and include also the maximal outer-planar graphs. These graphs can be used to model series and parallel electric circuits.Planar 3-trees are also known as Apollonian networks [2].A k -path is a k -tree with maximum degree 2 k , where for each integer j , k ≤ j < k , thereexists a unique pair of vertices, u and v , such that deg( u ) = deg( v ) = j . On the otherhand, regards to the recursive construction of k -trees, a k -path G can be considered as agraph with vertex set V ( G ) = { v , v , . . . , v n } and edge set E ( G ) = { v i v j : | i − j | ≤ k } . For instance, two different representations of a 2-path G with seven vertices v , . . . , v areshown in Figure 1. 2 v v v v v v v v v v v v v Figure 1: Two different representations of a 2-path.In this paper, we show that the metric dimension of each k -path (as a generalization of apath) is k . Whereas, there are some examples of 2-trees with metric dimension two thatare not 2-path. This fact motivates us to study the structure of 2-dimensional 2-trees. Asa main result, we characterize the class of all 2-trees with metric dimension two. In this section, we first prove that the metric dimension of each k -path is k . Then, weintroduce a class of graphs which shows that the inverse of this fact is not true in general.Later on, we concern on the case k = 2 and toward to investigating all 2-trees with metricdimension two, we construct a family F of 2-trees with metric dimension two. Finally, asthe main result, we prove that the metric dimension of a 2-tree G is two if and only if G belongs to F . Theorem 2.1 If G is a k -path, then dim M ( G ) = k . Proof.
Let G be a k -path with vertex set V ( G ) = { v , v , . . . , v n } and edge set E ( G ) = { v i v j : | i − j | ≤ k } . Therefore, the distance between two vertices v r and v s in G is givenby d ( v r , v s ) = l | r − s | k m .At first, let W = { v , v , . . . , v k } and v i , v j be two distinct vertices of G with k < i < j .By the division algorithm, there exist integers r and s such that i = rk + s , 1 ≤ s ≤ k .Thus, we have d ( v i , v s ) = (cid:24) | i − s | k (cid:25) = (cid:24) rkk (cid:25) = r, and d ( v j , v s ) = (cid:24) | j − s | k (cid:25) = (cid:24) rk + ( j − i ) k (cid:25) = r + (cid:24) j − ik (cid:25) ≥ r + 1 . This means W is a resolving set for G . Hence, dim M ( G ) ≤ | W | = k .Now, we show that dim M ( G ) ≥ k . Let W be a basis of the k -path G , and let X = { v , v , . . . , v k +1 } . Assume that | W ∩ X | = s and X \ W = { v i , v i , . . . , v i k +1 − s } , where1 ≤ i < i < · · · < i k +1 − s ≤ k + 1. For convince, let X ′ = { x , x , . . . , x k +1 − s } , where3 r = v i r , for each r , 1 ≤ r ≤ k + 1 − s . Since each vertex v i of the k -path G is adjacent tothe next k consecutive vertices { v i +1 , . . . , v i + k } , the induced subgraph on X is a ( k + 1)-clique. Each vertex in W ∩ X is adjacent to each vertex in X ′ . Thus, each pair of verticesin X ′ should be resolved by some element of W \ X . Assume that W ′ = { w , w , . . . , w t } is a minimum subset of W \ X which resolves vertices in X ′ . Thus, for each w j ∈ W ′ thereexists { x r , x s } ⊆ X ′ such that d ( w j , x r ) = d ( w j , x s ). For each j , 1 ≤ j ≤ t , let r j = min { r : d ( w j , x r ) = d ( w j , x r +1 ) } , and, let A j = { x , x , . . . , x r j } , B j = { x r j +1 , x r j +2 , . . . , x k +1 − s } . Note that A j ∪ B j = X ′ , A j ∩ B j = ∅ , x ∈ A j and x k +1 − s ∈ B j . Also, the structure of G implies that d ( w j , x ) = d ( w j , x ) = · · · = d ( w j , x r j ) , and d ( w j , x r j +1 ) = d ( w j , x r j +2 ) = · · · = d ( w j , x k +1 − s ) . Since W ′ has the minimum size, for each 1 ≤ j < j ′ ≤ t we have A j = A j ′ (otherwise, w j and w j ′ resolve the same pair of vertices in X ′ ) and hence, | A j | 6 = | A j ′ | . Moreover,for each r , 1 ≤ r ≤ k − s , there exists w j ∈ W ′ such that d ( w j , x r ) = d ( w j , x r +1 ) whichimplies | A j | = r . Therefore, t = |{| A | , | A | , . . . , | A t |}| = |{ , , . . . , k − s }| = k − s. Hence, | W | = | W \ X | + | W ∩ X | ≥ | W ′ | + s = ( k − s ) + s = k, which completes the proof. Definition 2.2
Let G and H be two -trees. We say that H is a branch in G on { u, v } ,for convenience say a ( u, v ) -branch, if V ( H ) ∩ V ( G ) = { u, v } , where uv is an edge of G belonging to only one of the triangles in H . The length of a branch in a -tree is thenumber of it’s triangles, which is equal to the number of vertices of branch minus . Acane is a -path with a branch of length one on a specific edge as shown in Figure 2. · · · Figure 2: A cane.In the following proposition, we provide some 2-trees with metric dimension two otherthan 2-paths.
Proposition 2.3 If G is a -tree of metric dimension two with a basis whose elementsare adjacent, then G is a -path or a cane. (1 , b (1 , ... (2 , t , t + 1) ( t , t )(a) a ... (b) b ... a b (c) a (1 , b (1 , ... ( t , t + 1) ( t , t )(d)Figure 3: The possible cases for basis { a, b } in 2-tree G Proof.
We prove the statement by induction on n , the order of G . If n = 3, then G = K and the statement holds. Let G be a 2-tree of order n > B = { a, b } , suchthat d ( a, b ) = 1. Since each 2-tree of order greater than three has two non-adjacent verticesof degree two, there exists a vertex x ∈ V ( G ) \ B of degree two. Moreover, B is a basisfor G \ { x } .Now, by the induction hypothesis, G \ { x } is a path or a cane and by Theorem 1.1 (2),the degrees of a and b are at most three. Therefore, B = { a, b } is one of the possible casesshown in Figure 3. Note that dashed edges could be absent. It can be checked that incases (b) and (c) the bold vertices get the same metric representation with respect to B .Thus, B is one of the cases (a) or (d), where the metric representations of vertices aredenoted in Figure 3.Regards to the metric representation of vertices in G , x could be adjacent to the verticesby metric representation ( t, t + 1) and ( t, t ) (in the case of not existence of dashed edges( t − , t ) and ( t, t )) and in the case (d) to the vertices by metric representation (1 ,
0) and(1 ,
1) as well. This concludes that G is also a path or a cane.The above proposition shows that the inverse of Theorem 2.1 is not true. Later on, wefocus on the case k = 2 and construct the family F of all 2-trees with metric dimensiontwo.Let F be the family of 2-trees, where each member G of F consists of a 2-tree G andsome branches on it that, in the case of existence, satisfying the following conditions.1. G is a 2-path or a 2-tree that is obtained by identifying two specific edges of twodisjoint 2-paths as shown in Figure 4.2. On every edge there is at most one branch.5. G avoids any ( a i , a i +1 )-branch.4. Each branch is either a 2-path or a cane.5. In each ( a i , b i )-branch the degree of a i is two.6. If G is as the graph depicted in Figure 4( b ), then G avoids any ( a m , x )-branch.7. G contains at most one branch on the edges of the triangle containing b i b i +1 in G .8. The degree of each b i in G is at most 7.9. G has at most one branch of length greater than one on the edges of the trianglecontaining a i a i +1 in G .10. If G is of the form of Figure 4( b ), then ( b m − , b m )-branch and ( b m , b m +1 )-branchare 2-path and at most one of them is of length more than one.11. For every i , 2 ≤ i ≤ k −
1, at most one of the ( b i − , b i )-branches and ( b i , b i +1 )-branches is a cane.12. All ( a i , b i )-branches, ( a i , b i +1 )-branches and ( a i , b i − )-branches are 2-path. a a a a k − a k b b b b k − b k · · · ( a ) a a a a m − a m − b b b b m − b m − · · · a m a m +1 a m +2 a k − a k b m b m +1 b m +2 b k − b k · · · ( b )Figure 4: Two different forms of G . Theorem 2.4 If G ∈ F , then dim M ( G ) = 2 . Proof.
Let G ∈ F . Through the proof all of notations are the same as those which areused to introduce the family F and G in Figure 4. Since G is not a path, dim M ( G ) ≥ W = { a , a k } . We show in both possible cases for G that W is a resolving set for G and hence, dim M ( G ) = 2. 6 ase 1. G is a 2-path as shown in Figure 4(a).The metric representation of the vertices { a , a , . . . , a k , b , b , . . . , b k } are as follows. r ( a i | W ) = ( i − , k − i ) , ≤ i ≤ k,r ( b | W ) = (1 , k ) ,r ( b j | W ) = ( j − , k − j + 1) , ≤ j ≤ k. Thus, different vertices of G have different metric representations. Moreover, note that { d − d : ( d , d ) = r ( a i | W ) , ≤ i ≤ k } = { − k, − k, − k, . . . , i − k − , . . . , k − , k − } , and { d − d : ( d , d ) = r ( b i | W ) , ≤ i ≤ k } = { − k, − k, − k, . . . , i − k − , . . . , k − , k − } . If G = G , then we are done. Suppose that G = G and let H be a branch of G on anedge e of G . Regards to the structures of graphs in F , we consider the following differentpossibilities. • H is a branch on the vertical edge e = a i b i , 2 ≤ i ≤ k − F , H is a 2-path and deg H ( a i ) = 2. Let V ( H ) = { x , x , . . . , x t } where x = a i , x = b i , and E ( H ) = { x r x s : | r − s | ≤ } . If j isodd, then d ( x j , a ) = d ( x j , a i ) + d ( a i , a ) and d ( x j , a k ) = d ( x j , a i ) + d ( a i , a k ). If j iseven, then d ( x j , a ) = d ( x j , b i ) + d ( b i , a ) and d ( x j , a k ) = d ( x j , b i ) + d ( b i , a k ). Hence,we have r ( x j | W ) = ( ( i − ⌊ j ⌋ , k − i + ⌊ j ⌋ ) j is odd( i − ⌊ j ⌋ − , k − i + ⌊ j ⌋ ) j is even . Moreover, note that { d − d : ( d , d ) = r ( x j | W ) , ≤ j ≤ t } = { i − k − , i − k − } . • H is a branch on the oblique edge e = a i b i +1 , 2 ≤ i ≤ k − F , H is a 2-path and deg H ( a i ) = 2. Let V ( H ) = { x , x , . . . , x t } where x = a i , x = b i +1 , and E ( H ) = { x r x s : | r − s | ≤ } . If j is odd, then d ( x j , a ) = d ( x j , a i ) + d ( a i , a ) and d ( x j , a k ) = d ( x j , a i ) + d ( a i , a k ). If j is even, then d ( x j , a ) = d ( x j , b i +1 ) + d ( b i +1 , a ) and d ( x j , a k ) = d ( x j , b i +1 ) + d ( b i +1 , a k ). Hence,we have r ( x j | W ) = ( ( i − ⌊ j ⌋ , k − i + ⌊ j ⌋ ) j is odd( i − ⌊ j ⌋ , k − i + ⌊ j ⌋ − j is even . Moreover, note that { d − d : ( d , d ) = r ( x j | W ) , ≤ j ≤ t } = { i − k − , i − k } . H is a branch on the horizontal edge e = b i b i +1 , 1 ≤ i ≤ k − F , H is either a 2-path or a cane. Generally, assume that { x , x , . . . , x t } ⊆ V ( H ) ⊆ { x , x , . . . , x t } ∪ { x } , where the induced subgraph of H on { x , x , . . . , x t } is a 2-path with the edge set { x r x s : | r − s | ≤ } . We consider two different possibilities.a) x = b i , x = b i +1 . Hence, if H is a cane, then we have N H ( x ) = { b i , x } .Similar to the previous cases, we have r ( x | W ) = ( i − , k − i + 1) ,r ( x j | W ) = ( ( i − ⌊ j ⌋ , k − i + ⌊ j ⌋ ) j ≥ i − ⌊ j ⌋ , k − i + ⌊ j ⌋ − j is even . Also, if H is a cane, then r ( x | W ) = ( i − , k − i + 2).b) x = b i +1 , x = b i . Hence, if H is a cane, then we have N H ( x ) = { b i +1 , x } .Similarly, we have r ( x | W ) = ( i − , k − i ) ,r ( x j | W ) = ( ( i − ⌊ j ⌋ , k − i + ⌊ j ⌋ ) j is odd( i − ⌊ j ⌋ − , k − i + ⌊ j ⌋ ) j is even . Also, if H is a cane, then r ( x | W ) = ( i − , k − i + 1).Note that in both states (and regardless of being a 2-path or a cane), we have { d − d : ( d , d ) = r ( v | W ) , v ∈ V ( H ) } = { i − k − , i − k − , i − k } . Therefore, in all the above cases, distinct vertices of H have different metric representa-tions. Also, the metric representation of the vertices in V ( H ) are different from the metricrepresentations of the vertices in V ( G ) \ { x, y } , where H is a ( x, y )-branch. Moreover,using the subtraction value of two coordinates in the metric representation of each vertex,it is easy to check that vertices of different (possible) branches on G (satisfying the con-ditions mentioned in the definition of F ) have different metric representations. Thus, inthis case W is a resolving set for G . Case 2. G is a 2-tree of the form Figure 4(b).The metric representation of the vertices { a , a , . . . , a m , . . . , a k } ∪ { b , b , . . . , b m , . . . , b k } are as follows. r ( a i | W ) = ( i − , k − i ) , ≤ i ≤ k,r ( b j | W ) = ( j, k − j ) 1 ≤ j ≤ m − m, k − m + 1) j = m ( j − , k − j + 1) m + 1 ≤ j ≤ k. G have different metric representations. Moreover, notethat { d − d : ( d , d ) = r ( a i | W ) , ≤ i ≤ k } = { − k, − k, − k, . . . , m − k − , m − k − , m − k + 1 , . . . , k − , k − } , and { d − d : ( d , d ) = r ( b j | W ) , ≤ j ≤ k } = { − k, − k, − k, . . . , m − k − , m − k − , m − k, . . . , k − , k − } . If G = G , then we are done. Hence, suppose that G = G and let H be a branch of G on an edge e of G . Again, using the possible structures of H according to the definitionof F , we consider the following different cases. • H is a branch on the vertical edge e = a i b i , 2 ≤ i ≤ m − F , H is a 2-path and deg H ( a i ) = 2. Let V ( H ) = { x , x , . . . , x t } where x = a i , x = b i , and E ( H ) = { x r x s : | r − s | ≤ } . It isstraightforward to check that r ( x j | W ) = ( ( i − ⌊ j ⌋ , k − i + ⌊ j ⌋ ) j is odd( i + ⌊ j ⌋ − , k − i + ⌊ j ⌋ − j is even . Moreover, note that { d − d : ( d , d ) = r ( x j | W ) , ≤ j ≤ t } = { i − k − , i − k } . • H is a branch on the vertical edge e = a i b i , m + 1 ≤ i ≤ k − F , H is a 2-path and deg H ( a i ) = 2. Let V ( H ) = { x , x , . . . , x t } where x = a i , x = b i , and E ( H ) = { x r x s : | r − s | ≤ } . We have r ( x j | W ) = ( ( i − ⌊ j ⌋ , k − i + ⌊ j ⌋ ) j is odd( i + ⌊ j ⌋ − , k − i + ⌊ j ⌋ ) j is even . Moreover, note that { d − d : ( d , d ) = r ( x j | W ) , ≤ j ≤ t } = { i − k − , i − k − } . • H is a branch on the oblique edge e = a i b i − , 2 ≤ i ≤ m − G ∈ F , H is a 2-path and deg H ( a i ) = 2. Let V ( H ) = { x , x , . . . , x t } where x = a i , x = b i − , and E ( H ) = { x r x s : | r − s | ≤ } . We have r ( x j | W ) = ( ( i − ⌊ j ⌋ , k − i + ⌊ j ⌋ ) j is odd( i + ⌊ j ⌋ − , k − i + ⌊ j ⌋ ) j is even . Moreover, { d − d : ( d , d ) = r ( x j | W ) , ≤ j ≤ t } = { i − k − , i − k − } . H is a branch on the oblique edge e = a i b i +1 , m + 1 ≤ i ≤ k − H is a 2-path and deg H ( a i ) = 2. Let V ( H ) = { x , x , . . . , x t } where x = a i , x = b i +1 , and E ( H ) = { x r x s : | r − s | ≤ } . Similarly, it can be easilychecked that r ( x j | W ) = ( ( i − ⌊ j ⌋ , k − i + ⌊ j ⌋ ) j is odd( i + ⌊ j ⌋ − , k − i + ⌊ j ⌋ − j is even . Moreover, note that { d − d : ( d , d ) = r ( x j | W ) , ≤ j ≤ t } = { i − k − , i − k } . • H is a branch on the horizontal edge e = b i b i +1 , 1 ≤ i ≤ m − F , H is either a 2-path or a cane. Generally, assume that { x , x , . . . , x t } ⊆ V ( H ) ⊆ { x , x , . . . , x t } ∪ { x } , where the induced subgraph of H on { x , x , . . . , x t } is a 2-path with the edge set { x r x s : | r − s | ≤ } . We consider two different possibilities.a) x = b i , x = b i +1 . Hence, if H is a cane, then we have N H ( x ) = { b i , x } .Similar to the previous cases, we have r ( x | W ) = ( i, k − i ) ,r ( x j | W ) = ( ( i + ⌊ j ⌋ , k − i + ⌊ j ⌋ − j ≥ i + ⌊ j ⌋ , k − i + ⌊ j ⌋ − j is even . Also, if H is a cane, then r ( x | W ) = ( i + 1 , k − i + 1).b) x = b i +1 , x = b i . Hence, if H is a cane, then we have N H ( x ) = { b i +1 , x } .Similarly, we have r ( x | W ) = ( i + 1 , k − i − ,r ( x j | W ) = ( ( i + ⌊ j ⌋ , k − i + ⌊ j ⌋ − j ≥ i + ⌊ j ⌋ − , k − i + ⌊ j ⌋ −
1) is even . Also, if H is a cane, then r ( x | W ) = ( i + 2 , k − i ).Note that in the both states (and regardless of being a 2-path or a cane) we have { d − d : ( d , d ) = r ( v | W ) , v ∈ V ( H ) } = { i − k, i − k + 1 , i − k + 2 } . • H is a branch on the horizontal edge e = b m − b m .By the definition of F , H is a 2-path and deg H ( b m − ) = 2. Let V ( H ) = { x , x , . . . , x t } where x = b m − , x = b m , and E ( H ) = { x r x s : | r − s | ≤ } . We have r ( x j | W ) = ( ( m + ⌊ j ⌋ − , k − m + ⌊ j ⌋ + 1) j is odd( m + ⌊ j ⌋ − , k − m + ⌊ j ⌋ ) j is even . { d − d : ( d , d ) = r ( x j | W ) , ≤ j ≤ t } = { m − k − , m − k − } . • H is a branch on the horizontal edge e = b m b m +1 .By the definition of F , H is a 2-path and deg H ( b m +1 ) = 2. Let V ( H ) = { x , x , . . . , x t } where x = b m +1 , x = b m , and E ( H ) = { x r x s : | r − s | ≤ } . We have r ( x j | W ) = ( ( m + ⌊ j ⌋ , k − m + ⌊ j ⌋ ) j is odd( m + ⌊ j ⌋ − , k − m + ⌊ j ⌋ ) j even . Moreover, note that { d − d : ( d , d ) = r ( x j | W ) , ≤ j ≤ t } = { m − k − , m − k } . • H is a branch on the horizontal edge e = b i b i +1 , m + 1 ≤ i ≤ k − F , H is either a 2-path or a cane. Generally, assume that { x , x , . . . , x t } ⊆ V ( H ) ⊆ { x , x , . . . , x t } ∪ { x } , where the induced subgraph of H on { x , x , . . . , x t } is a 2-path with the edge set { x r x s : | r − s | ≤ } . Again, we consider two different possibilities.a) x = b i , x = b i +1 . Hence, if H is a cane and N H ( x ) = { b i , x } , then We have r ( x | W ) = ( i − , k − i + 1) ,r ( x j | W ) = ( ( i + ⌊ j ⌋ − , k − i + ⌊ j ⌋ ) j ≥ i + ⌊ j ⌋ − , k − i + ⌊ j ⌋ − j is even . Also, if H is a cane, then r ( x | W ) = ( i, k − i + 2).b) x = b i +1 , x = b i . Hence, if H is a cane, then we have N H ( x ) = { b i +1 , x } .Similarly, we have r ( x | W ) = ( i, k − i ) ,r ( x j | W ) = ( ( i + ⌊ j ⌋ − , k − i + ⌊ j ⌋ ) j ≥ i + ⌊ j ⌋ − , k − i + ⌊ j ⌋ ) j is even . Also, if H is a cane, then r ( x | W ) = ( i + 1 , k − i + 1).Note that in the both states (and regardless of being a 2-path or a cane) we have { d − d : ( d , d ) = r ( v | W ) , v ∈ V ( H ) } = { i − k − , i − k − , i − k } . H have different metric representations.Also, the metric representation of the vertices in V ( H ) are different from the metricrepresentations of the vertices in V ( G ) \ { x, y } , where H is a ( x, y )-branch. Moreover,using the subtraction value of two coordinates in the metric representation of each vertex,it is easy to check that vertices of different (possible) branches on G (satisfying theconditions mentioned in the definition of F ) have different metric representations. Thus,in this case W is a resolving set for G .To prove the converse of Theorem 2.4, we need the following lemma. Lemma 2.5
Let H be a { u, v } -branch of G and let { a, b } be a basis for G ∪ H . If { a, b } ∩ V ( H ) ⊆ { u, v } , then { u, v } is a metric basis for H . Proof.
Suppose on the contrary, there are two different vertices x and y in H such that d ( x, u ) = d ( y, u ) = r, d ( x, v ) = d ( y, v ) = s. Since H is a branch on { u, v } , each path connecting a vertex in H with a vertex in V ( G ) \ V ( H ) passes through u or v . Assume that d ( u, a ) = r , d ( v, a ) = s , d ( u, b ) = r , d ( v, b ) = s . Hence, d ( x, a ) = min { r + r , s + s } = d ( y, a ) , d ( x, b ) = min { r + r , s + s } = d ( y, b ) . This contradicts that { a, b } is a resolving set for G ∪ H .Now, we prove that every 2-dimensional 2-tree belongs to the family F . Theorem 2.6 If G is a -tree of metric dimension two, then G ∈ F . Proof.
Let G be a 2-tree and { a, b } be a basis of G . If d ( a, b ) = 1, then by Proposition 2.3, G is a 2-path or a cane which belongs to F . Thus, assume that d ( a, b ) > H be a minimal induced 2-connected subgraph of G as shown in Figure 5, containing a and b . Since the clique number of G is three, in each square exactly one of the dashed edgesare allowed. Moreover, by the minimality of H we have deg H ( a ) = deg H ( b ) = 2, where a ∈ { a , b } and b ∈ { a k , b k } . Hence, one of two vertices a , b or one of two vertices a k , b k may not exist. One can check that { a, b } 6 = { a , b k } and { a, b } 6 = { b , a k } , otherwise, twoneighbours of a or b get the same metric representation. Thus, by the symmetry, we mayassume { a, b } = { a , a k } . a a k b b k · · · Figure 5: A minimal induced 2-connected subgraph of G .12f ∆( H ) ≤
4, then H is a 2-path as shown in Figure 4(a). Otherwise ∆( H ) = 5. If thereexists a vertex b j of degree 5, then it can be easily checked that b j and a j have the samerepresentation with respect to { a , a k } . Also, existence of two vertices a i and a i ′ both ofdegree 5, i ≤ i ′ , implies that there exists some vertex b j , i ≤ j ≤ i ′ , of degree 5, which isimpossible. Thus, there exists a unique a i of degree 5. Therefore, H is the graph shown inFigure 4(b). Thus, H is a 2-path or a 2-tree obtained by identifying the specific edge, say a m b m , of two 2-paths (see Figure 4(b)), where B = { a , a k } . Thus, G satisfies property(1).Clearly, on every edge there is at most one branch; thus, property (2) follows. Also, G avoids any ( a i , a i +1 )-branch, because each vertex adjacent to both a i and a i +1 has thesame metric representation as b i or b i +1 . Thus, G contains only ( a i , b i )-branches, ( a i , b i +1 )-branches, ( a i +1 , b i )-branches or ( b i , b i +1 )-branches; which implies property (3). Moreover,by Proposition 2.3 and Lemma 2.5, each of these branches is a 2-path or a cane. Therefore,property (4) holds. Also, by Theorem 1.1, for every i , 1 ≤ i ≤ k , there is at most one( a i , x )-branch in G . Moreover, in each ( a i , b i )-branch the degree of a i is two, which showstrueness of property (5).To see property (6), first note that by property (3) there is no ( a m − , a m )-branch or( a m , a m +1 )-branch. Moreover, in each ( a m , x )-branch, for x ∈ { b m − , b m , b m +1 } , theunique neighbour of a m on the branch has the same metric representation as b m .To show that G has property (7), suppose that a triangle a i b i b i +1 has more than onebranch. By Theorem 1.1, at most one of ( a i , b i )-branch and ( a i , b i +1 )-branch exists. There-fore, b i b i +1 has a branch H and one of the edges a i b i or a i b i +1 has another branch H .Let x and y be the vertices of distance one from G on branches H and H , respectively.Hence, d ( a , x ) = d ( a , y ) = i and d ( a k , x ) = d ( a k , y ) = k − i + 1. That is, { a , a k } is nota basis of G , which is a contradiction. A similar reason works for triangle a i b i − b i . Hence, G has property (7).Let ( d , d ) be metric representation of b i . Then metric representations of each neighbourof b i which is out of G could be one of ( d + 1 , d + 1) , ( d + 1 , d ) or ( d , d + 1). Thus, b i has at most three neighbours out of G . Hence, the degree of b i in G is at most 7 thatis property (8).If there are two branches of length at least 2 on a triangle containing a i a i +1 , then the metricrepresentation of the second vertices on these branches are the same, a contradiction. Thus, G satisfies property (9).If H is a ( b m − , b m )-branch of cane type, then one can find two vertices in N G ( b m ) ∪ N G ( b m − ) with the same metric representation. A similar argument holds whenever H isa ( b m , b m +1 )-branch of cane type. If there is a ( b m − , b m )-branch, say H , and a ( b m , b m +1 )-branch, say H , both of length at least two, then b m has a neighbour in H with the samemetric representation as a neighbour of b m in H . Hence, property (10) holds.Suppose that two branches on ( b i − , b i ) and ( b i , b i +1 ) are canes. In this case, it can bechecked that in the set of neighbours of b i in these branches there are two vertices withthe same metric representation. Thus, G satisfies property (11).13sing Theorem 1.1 the degree of each a i in G , 1 < i < n , is at most five. Note thatdeg( a i ) ∈ { , } . Now suppose that H is a branch on the edge { a i , b i } , { a i , b i +1 } or { a i , b i − } . If H is a cane, then deg G ( a i ) ≥ b i − , b i or b i +1 in H getthe same metric representation, which both are contradictions. Thus, each branch on theedge { a i , b i − } , { a i , b i } or { a i , b i +1 } is a 2-path and G satisfies property (12). References [1]
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