The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes

For a setW = {w1, w2, . . . , wk} of vertices and a vertex V of a connected graph G, the multirepresentation of V with respect toW is the k-multiset mr(V | W) = {d(V, w1), d(V, w2), . . . , d(V, wk)}, where d(V, wi) is the distance between the vertices V and wi for i = 1, 2, . . . , k. The set W is a multiresolving set of G if every two distinct vertices of G have distinct multirepresentations with respect to W. The minimum cardinality of a multiresolving set of G is the multidimension dimM(G) of G. It is shown that, for every pair k, n of integers with k ≥ 3 and n ≥ 3(k − 1), there is a connected graph G of order n with dimM(G) = k. For a multiset {a1, a2, . . . , ak} and an integer c, we define {a1, a2, . . . , ak} + {c, c, . . . , c} = {a1 + c, a2 + c, . . . , ak + c}. A multisimilar equivalence relation RW on V(G) with respect to W is defined by u RW V if mr(u | W) = mr(V | W) + {cW(u, V), cW(u, V), . . . , cW(u, V)} for some integer cW(u, V). We study the relationship between the elements in multirepresentations of vertices that belong to the same multisimilar equivalence class and also establish the upper bound for the cardinality of a multisimilar equivalence class. Moreover, a multiresolving set with prescribed multisimilar equivalence classes is presented.


Introduction
The distance (, V) between two vertices  and V in a connected graph  is the length of a shortest  − V path in .For an ordered set  = { 1 ,  2 , . . .,   } ⊆ () and a vertex V of , the -vector  (V | ) = ( (V,  1 ) ,  (V,  2 ) , . . .,  (V,   )) is referred to as the representation of V with respect to .The ordered set  is called a resolving set of  if every two distinct vertices of  have distinct representations with respect to .A resolving set of a minimum cardinality is called a minimum resolving set or a basis of  and this cardinality is the dimension dim() of .
To illustrate these concepts, consider a connected graph  of Figure 1   Since there is no 1-element resolving set of , it follows that  1 is a basis of , and so dim() = 2.
The concepts of resolving sets and minimum resolving sets have previously appeared in [1][2][3][4].Slater in [3,4] introduced these ideas and used a locating set for what we have called a resolving set.He referred to the cardinality of a minimum resolving set in a connected graph as its locating number.He described the usefulness of these ideas when working with US sonar and coast guard LORAN (long range aids to navigation) stations.Harary and Melter [2] discovered these concepts independently as well but used the term metric dimension rather than locating number, the terminology that we have adopted.These concepts were rediscovered by Johnson [5] of the Pharmacia Company while attempting to develop a capability of large datasets of chemical graphs.More applications of these concepts to navigation of robots in networks and other areas are discussed in [6][7][8][9].
A multiset is a generalization of the concept of a set, which is like a set except that its members need not to be distinct.For example, the set {1, 1, 2} is the same as the set {1, 2} but not so for the multiset.The multiset  = {5, 5, 6, , , , , , , } has 10 elements of 4 different types: 2 of type 5, 1 of type 6, 4 of type , and 3 of type .So, the multiset is usually indicated by specifying the number of times different types of elements occur in it.Therefore, the multiset  can be written by  = {2 ⋅ 5, 1 ⋅ 6, 4 ⋅ , 3 ⋅ }.The numbers 2, 1, 4, and 3 are called the repetition numbers of the multiset .In particular, a set is a multiset having all repetition numbers equal to 1.
As described in [1], all connected graphs  contain an ordered set  such that each vertex of  is distinguished by a -vector, known as a representation, consisting of its distance from the vertices in .It may also occur that some graph contains a set   with property that the vertices of graph have uniquely distinct -multisets containing their distances from each of the vertices in   .The goal of this paper is to study the existence of such a set of connected graphs.
For a set  = { 1 ,  2 , . . .,   } of vertices and a vertex V of a connected graph , we refer to the -multiset as the multirepresentation of V with respect to .The set  is called a multiresolving set of  if every two distinct vertices have distinct multirepresentations with respect to .A multiresolving set of a minimum cardinality is called a minimum multiresolving set or a multibasis of  and this cardinality is the multidimension dim  () of .
For example, consider a connected graph  of Figure 1.As we know  1 = {, } is a basis of .However,  1 is not a multiresolving set of  since ( |  1 ) = {2, 3} = (V |  1 ).In fact, the set  2 = {, , } is a multiresolving set of  with the following multirepresentations of the vertices of  with respect to  2 : It is routine to verify that there are no 1-element and 2element multiresolving sets of .Hence,  2 is a multibasis of , and so dim  () = 3.
Not all connected graphs have a multiresolving set and also dim  () is not defined for all connected graphs .For example, the complete graph  3 has no multiresolving set.Thus, dim  ( 3 ) is not defined.However, if  is a connected graph of order , for which dim  () is defined, and then every multiresolving set of  is a resolving set of , and so For every set  of vertices of a connected graph , the vertices of  whose multirepresentations with respect to  contain 0 are vertices in .On the other hand, the multirepresentations of vertices of  which do not belong to  have elements, all of which are positive.In fact, to determine whether a set  is a multiresolving set of , the vertex set () can be partitioned into  and () −  to examine whether the vertices in each subset have distinct multirepresentations with respect to .
The multiresolving set of a connected graph was introduced by Saenpholphat [10] who showed that there is no connected graph  such that dim  () = 2.Moreover, the multidimensions of complete graphs, paths, cycles, and bipartite graphs were determined.Simanjuntak, Vetrík, and Mulia [11] discovered this concept independently and used a notation () for a multidimension of a connected graph .

The Multidimension of a Connected Graph
Two vertices  and V of a connected graph  are distancesimilar if (, ) = (V, ) for all  ∈ () − {, V}.Certainly, distance similarity in  is an equivalence relation on ().For example, consider a complete bipartite graph  , with partite sets  and .Every pair of vertices in the same partite set are distance-similar.Then the distance-similar equivalence classes in  , are its partite sets  and .The following results were obtained in [10] showing the usefulness of the distance-similar equivalence class to determine the multidimensions of connected graphs.
Theorem 1 (see [10]).Let  be a connected graph such that dim  () is defined.If  is a distance-similar equivalence class in  with || = 2, then every multiresolving set of  contains exactly one vertex of .
Theorem 2 (see [10]).If  is a distance-similar equivalence class in a connected graph  with || ≥ 3, then dim  () is not defined.
It was shown in [10,11] that a path is the only one of connected graphs with multidimension 1, and any multiresolving sets of a connected graph cannot contain only two vertices.We state these results in the following theorems.
Last, we are able to determine all pairs ,  of positive integers with  ≥ 3 and  ≥ 3( − 1) which are realizable as the multidimension and the order of some connected graph.In order to do this, we present an additional notation.For integers  and , let [, ] be a multiset such that Such a multiset is referred to as a consecutive multiset of integers  and .
Theorem 5.For every pair ,  of integers with  ≥ 3 and  ≥ 3( − 1), there is a connected graph  of order  with dim  () = .
We consider two cases.

Multisimilar Equivalence Relation
In this section, we investigate another equivalence relation on a vertex set of a connected graph.First, we need some additional definitions and notations.Let  = {{   ( The function  is called a multisimilar function of     or a multisimilar function if there is no ambiguity.Consequently, it is not surprising that an inverse function  −1 is also multisimilar function of V    with a multisimilar constant   (V, ) = −  (, V).
To illustrate these concepts, consider a vertex  in a connected graph  of Figure 1 The example just described shows an important point that the multisimilar function of any two vertices in the same multisimilar equivalence class with respect to a set  is not necessarily unique.More generally, for a vertex  and a set  of vertices of a connected graph , let ( | ) = { 1 ⋅  1 ,  2 ⋅  2 , . . .,   ⋅   }, where  1 <  2 < ⋅ ⋅ ⋅ <   and   is a repetition number of type   for each  with 1 ≤  ≤ .If  ∈ [V]  , where V ∈ (), then it follows by ( 13) and ( 14) that, for each type of ( | ), there is a corresponding type of (V | ) such that their repetition numbers are equal.Therefore, we may assume that International Journal of Mathematics and Mathematical Sciences 5   } and   = { ∈  | (V, ) =   }.Then the types of ( | ) partition  into  sets  1 ,  2 , . . .,   .On the other hand,  is also partitioned into  sets  1 ,  2 , . . .,   depending on the types of (V | ).Hence, the multisimilar function  of    V has the property that, for every vertex  ∈   , there is a vertex   ∈   such that where 1 ≤  ≤ .Indeed, there are  1 ! 2 !⋅ ⋅ ⋅   !distinct multisimilar functions of    V.These observations yield the following result.where   (, V) is a multisimilar constant.Thus, (i) holds.For (ii), the statement may be proven in the same way as (i), and therefore such proof is omitted.
Next, we are prepared to establish the upper bound for the cardinality of a multisimilar equivalence class of a vertex in a connected graph.To show this, let us present a useful proposition as follows.We can show that the upper bound in Theorem 9 is sharp.Consider the path   = (V 1 , V 2 , . . ., V  ).We have that diam(  ) = −1 and the set  = {V 1 } is a multiresolving set of   .Thus, [V 1 ]  contains all vertices of   , and so In the last result, we describe the properties of a multisimilar equivalence classes with respect to a set of vertices.

Figure 2 :
Figure 2: A connected graph  in Case 1.

Proposition 8 .
Let  be a set of vertices of a connected graph  and let  and V be vertices of  such that  ∈ [V]  .Then ( | ) and (V | ) have the same minimum (or maximum) element if and only if ( | ) = (V | ).

Theorem 9 .
If  is a multiresolving set of a connected graph , then the cardinality of multisimilar equivalence class of each vertex of  with respect to  is at most diam() + 1.Proof.Assume, to the contrary, that there is a vertex V of  such that [V]  has the cardinality at least diam() + 2. Since the minimum elements of multirepresentations of vertices in [V]  with respect to  have at most diam() + 1 distinct values, there are at least two vertices  and  in [V]  having the same value of the minimum element of ( | ) and ( | ).It follows by Proposition 8 that ( | ) = ( | ), contradicting the fact that  is a multiresolving set of .