Abstract
For a set of vertices and a vertex of a connected graph , the multirepresentation of with respect to is the -multiset where is the distance between the vertices and for . The set is a multiresolving set of if every two distinct vertices of have distinct multirepresentations with respect to . The minimum cardinality of a multiresolving set of is the multidimension of . It is shown that, for every pair of integers with and , there is a connected graph of order with . For a multiset and an integer , we define . A multisimilar equivalence relation on with respect to is defined by if for some integer . 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.
1. Introduction
The distance between two vertices and in a connected graph is the length of a shortest path in . For an ordered set and a vertex of , the -vector is referred to as the representation of 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 of .
To illustrate these concepts, consider a connected graph of Figure 1 with . Considering the ordered set , there are six representations of the vertices of with respect to :Since there is no 1-element resolving set of , it follows that is a basis of , and so .
The concepts of resolving sets and minimum resolving sets have previously appeared in [1–4]. 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 [6–9].
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 is the same as the set but not so for the multiset. The multiset has 10 elements of 4 different types: 2 of type , 1 of type , 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 . The numbers , 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 of vertices and a vertex of a connected graph , we refer to the -multiset as the multirepresentation of 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 of .
For example, consider a connected graph of Figure 1. As we know is a basis of . However, is not a multiresolving set of since . In fact, the set is a multiresolving set of with the following multirepresentations of the vertices of with respect to :It is routine to verify that there are no 1-element and 2-element multiresolving sets of . Hence, is a multibasis of , and so .
Not all connected graphs have a multiresolving set and also is not defined for all connected graphs . For example, the complete graph has no multiresolving set. Thus, is not defined. However, if is a connected graph of order , for which 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 . 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 .
2. The Multidimension of a Connected Graph
Two vertices and of a connected graph are distance-similar if for all . 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 is defined. If is a distance-similar equivalence class in with , 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 , then is not defined.
It was shown in [10, 11] that a path is the only one of connected graphs with multidimension , and any multiresolving sets of a connected graph cannot contain only two vertices. We state these results in the following theorems.
Theorem 3 (see [10, 11]). Let be a connected graph. Then if and only if , the path of order .
Theorem 4 (see [10, 11]). A connected graph has no multiresolving set of cardinality
Last, we are able to determine all pairs of positive integers with and 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 thatSuch a multiset is referred to as a consecutive multiset of integers and .
Theorem 5. For every pair of integers with and , there is a connected graph of order with .
Proof. Let and be integers with and . We consider two cases.
Case 1 (). Let be a graph obtained from the path by adding the vertices and for and joining and to , as it is shown in Figure 2. Then the order of is . First, we claim that there is no multiresolving set of with cardinality at most . Assume, to the contrary, that there is a multiresolving set of such that . Since a set for is a distance-similar equivalence class in , it follows by Theorem 1 that contains exactly one vertex of . Without loss of generality, let for . Thus, . Since for all , it follows that and so a set is not a multiresolving set of , thereby producing a contradiction. Hence, . Next, we claim that a set is a multiresolving set of . For a vertex , the multirepresentation of with respect to isFor , the multirepresentation of with respect to isFor , the multirepresentation of with respect to isTherefore, is a multiresolving set of with . Hence, .
Case 2 (). Let be a graph obtained from the graph in Case 1 by adding the path and joining to and , as it is shown in Figure 3. By a similar argument to the one used in Case 1, it is shown that there is no multiresolving set of with . We claim that a set is a multiresolving set of . For vertices in , their multirepresentations with respect to are the same as in Case 1. For , the multirepresentation of with respect to is Hence, is a multiresolving set of with , and so .
3. 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 be a collection of multisets. For an integer , we define where . Let be a set of vertices of a connected graph and let and be vertices of . A multisimilar relation with respect to on a vertex set is defined by if there is an integer such thatAn integer satisfying (12) is called a multisimilar constant of or simply a multisimilar constant. Clearly, is an equivalence relation on . For each vertex in , let denote the multisimilar equivalence class of with respect to . Thenwhere is a multisimilar constant. Observe that if , then there is a multisimilar constant with a property that, for every vertex , there is a corresponding vertex such thatWith this observation, we may as well say that if and only if there are multisimilar constant and a bijective function on defined as 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 is also multisimilar function of with a multisimilar constant .
To illustrate these concepts, consider a vertex in a connected graph of Figure 1 and the set . There is only one vertex in such that is related to by a multisimilar relation with a multisimilar constant ; that is, Therefore, . Thus, a multisimilar function of is defined by Moreover, there is another multisimilar function of ; that is, 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 , where and is a repetition number of type for each with . If , where , then it follows by (13) and (14) that, for each type of , there is a corresponding type of such that their repetition numbers are equal. Therefore, we may assume that , where . For each integer with , let and . Then the types of partition into sets . On the other hand, is also partitioned into sets depending on the types of . Hence, the multisimilar function of has the property that, for every vertex , there is a vertex such that where . Indeed, there are distinct multisimilar functions of . These observations yield the following result.
Theorem 6. Let be a set of vertices of a connected graph and let and be vertices of such that . Suppose that , where and is a repetition number of type for each with . Then (i) for some integers with ,(ii)there is a multisimilar function of such that , where and for each with ,(iii)there are distinct multisimilar functions of .
By Theorem 6, the following result is obtained.
Corollary 7. Let be a set of vertices of a connected graph and let and be vertices of such that with a multisimilar constant . Then (i)if and are the maximum elements of and , respectively, then ,(ii)if and are the minimum elements of and , respectively, then .
Proof. Suppose that . Let and , where and . Since and are the maximum elements of and , respectively, there are vertices and in such that and . It follows by Theorem 6 that there is a multisimilar function of such that . Then , where 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.
Proposition 8. Let be a set of vertices of a connected graph and let and be vertices of such that . Then and have the same minimum (or maximum) element if and only if .
Proof. If , then the minimum (and maximum) elements of and are the same. For the converse, assume that and are the minimum elements of and , respectively, such that . Since , there is a multisimilar constant such that By Corollary 7 (ii), it follows that . Thus, . Hence, . Similarly, if and have the same maximum element, then .
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 .
Proof. Assume, to the contrary, that there is a vertex of such that has the cardinality at least . Since the minimum elements of multirepresentations of vertices in with respect to have at most distinct values, there are at least two vertices and in 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 .
We can show that the upper bound in Theorem 9 is sharp. Consider the path . We have that and the set is a multiresolving set of . Thus, 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.
Theorem 10. Let and be vertices of a connected graph and let be a set of vertices of . Then (i)if , then for all and ,(ii)if for all , then is a multiresolving set of .
Proof. (i) Assume, to the contrary, that there exist two distinct vertices and such that . Then there are multisimilar constants and such that and . Therefore, Thus, Hence, belongs to , which is a contradiction.
(ii) Assume, to the contrary, that is not a multiresolving set of . Then there exist two distinct vertices and such that . Hence, belongs to , producing a contradiction.
4. Final Remarks
The complete graph is only one graph that its dimension is but not so for multidimensions. It follows by [10, 11] that the multidimension of complete graph is not defined. Thus, (5) leads us to the conjecture: If is a connected graph such that is defined, then .
Data Availability
No data sharing was used to support this study as no datasets were generated or analyzed during the current study. Other data sources are referenced throughout the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was funded by the Faculty of Science, Srinakharinwirot University.