For a set W=w1,w2,…,wk of vertices and a vertex v of a connected graph G, the multirepresentation of v with respect to W is the k-multiset mr(v∣W)=dv,w1,dv,w2,…,dv,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 uRWv if mr(u∣W)=mrv∣W+cWu,v,cWu,v,…,cWu,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.
Srinakharinwirot University1. Introduction
The distanced(u,v) between two vertices u and v in a connected graph G is the length of a shortest u-v path in G. For an ordered set W=w1,w2,…,wk⊆V(G) and a vertex v of G, the k-vector (1)rv∣W=dv,w1,dv,w2,…,dv,wkis referred to as the representation ofv with respect to W. The ordered set W is called a resolving set of G if every two distinct vertices of G have distinct representations with respect to W. A resolving set of a minimum cardinality is called a minimum resolving set or a basis of G and this cardinality is the dimensiondim(G) of G.
To illustrate these concepts, consider a connected graph G of Figure 1 with V(G)=u,v,w,x,y,z. Considering the ordered set W1=w,z, there are six representations of the vertices of G with respect to W1:(2)ru∣W1=2,3,rv∣W1=3,2,rw∣W1=0,3,rx∣W1=1,2,ry∣W1=2,1,rz∣W1=3,0.Since there is no 1-element resolving set of G, it follows that W1 is a basis of G, and so dim(G)=2.
A connected graph G.
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 {1,1,2} is the same as the set {1,2} but not so for the multiset. The multiset M=5,5,6,a,a,a,a,b,b,b has 10 elements of 4 different types: 2 of type 5, 1 of type 6, 4 of type a, and 3 of type b. So, the multiset is usually indicated by specifying the number of times different types of elements occur in it. Therefore, the multiset M can be written by M=2·5,1·6,4·a,3·b. The numbers 2,1,4, and 3 are called the repetition numbers of the multiset M. In particular, a set is a multiset having all repetition numbers equal to 1.
As described in [1], all connected graphs G contain an ordered set W such that each vertex of G is distinguished by a k-vector, known as a representation, consisting of its distance from the vertices in W. It may also occur that some graph contains a set W′ with property that the vertices of graph have uniquely distinct k-multisets containing their distances from each of the vertices in W′. The goal of this paper is to study the existence of such a set of connected graphs.
For a set W=w1,w2,…,wk of vertices and a vertex v of a connected graph G, we refer to the k-multiset (3)mrv∣W=dv,w1,dv,w2,…,dv,wkas the multirepresentation of v with respect to W. The set W is called a multiresolving set of G if every two distinct vertices have distinct multirepresentations with respect to W. A multiresolving set of a minimum cardinality is called a minimum multiresolving set or a multibasis of G and this cardinality is the multidimension dimM(G) of G.
For example, consider a connected graph G of Figure 1. As we know W1=w,z is a basis of G. However, W1 is not a multiresolving set of G since mr(u∣W1)=2,3=mr(v∣W1). In fact, the set W2=w,x,z is a multiresolving set of G with the following multirepresentations of the vertices of G with respect to W2:(4)mru∣W2=1,2,3,mrv∣W2=2,2,3,mrw∣W2=0,1,3,mrx∣W2=0,1,2,mry∣W2=1,1,2,mrz∣W2=0,2,3.It is routine to verify that there are no 1-element and 2-element multiresolving sets of G. Hence, W2 is a multibasis of G, and so dimM(G)=3.
Not all connected graphs have a multiresolving set and also dimM(G) is not defined for all connected graphs G. For example, the complete graph K3 has no multiresolving set. Thus, dimM(K3) is not defined. However, if G is a connected graph of order n, for which dimM(G) is defined, and then every multiresolving set of G is a resolving set of G, and so(5)1≤dimG≤dimMG≤n.
For every set W of vertices of a connected graph G, the vertices of G whose multirepresentations with respect to W contain 0 are vertices in W. On the other hand, the multirepresentations of vertices of G which do not belong to W have elements, all of which are positive. In fact, to determine whether a set W is a multiresolving set of G, the vertex set V(G) can be partitioned into W and V(G)-W to examine whether the vertices in each subset have distinct multirepresentations with respect to W.
The multiresolving set of a connected graph was introduced by Saenpholphat [10] who showed that there is no connected graph G such that dimM(G)=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 md(G) for a multidimension of a connected graph G.
2. The Multidimension of a Connected Graph
Two vertices u and v of a connected graph G are distance-similar if d(u,x)=d(v,x) for all x∈V(G)-u,v. Certainly, distance similarity in G is an equivalence relation on V(G). For example, consider a complete bipartite graph Kr,s with partite sets U and V. Every pair of vertices in the same partite set are distance-similar. Then the distance-similar equivalence classes in Kr,s are its partite sets U and V. 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 G be a connected graph such that dimM(G) is defined. If U is a distance-similar equivalence class in G with U=2, then every multiresolving set of G contains exactly one vertex of U.
Theorem 2 (see [10]).
If U is a distance-similar equivalence class in a connected graph G with U≥3, then dimM(G) 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.
Theorem 3 (see [10, 11]).
Let G be a connected graph. Then dimM(G)=1 if and only if G=Pn, the path of order n.
Theorem 4 (see [10, 11]).
A connected graph has no multiresolving set of cardinality 2.
Last, we are able to determine all pairs k,n of positive integers with k≥3 and n≥3(k-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 a and b, let [a,b] be a multiset such that(6)a,b=a,a+1,…,b-1,bif a<baif a=b∅if a>b.Such a multiset is referred to as a consecutive multiset of integers a and b.
Theorem 5.
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.
Proof.
Let k and n be integers with k≥3 and n≥3(k-1). We consider two cases.
Case 1 (n=3(k-1)). Let G be a graph obtained from the path Pk-1=(u1,u2,…,uk-1) by adding the 2(k-1) vertices vi and wi for 1≤i≤k-1 and joining vi and wi to ui, as it is shown in Figure 2. Then the order of G is n=3(k-1). First, we claim that there is no multiresolving set of G with cardinality at most k-1. Assume, to the contrary, that there is a multiresolving set S of G such that S≤k-1. Since a set Vi=vi,wi for 1≤i≤k-1 is a distance-similar equivalence class in G, it follows by Theorem 1 that S contains exactly one vertex of Vi. Without loss of generality, let wi∈S for 1≤i≤k-1. Thus, S=k-1. Since d(w1,wi)=d(wk-1,wk-i) for all 1≤i≤k-1, it follows that mr(w1∣S)=mr(wk-1∣S) and so a set S=w1,w2,…,wk-1 is not a multiresolving set of G, thereby producing a contradiction. Hence, dimM(G)≥k. Next, we claim that a set W=w1,w2,…,wk-1∪u1 is a multiresolving set of G. For a vertex x∈W, the multirepresentation of x with respect to W is(7)mrx∣W=0,i∪3,i+1∪3,k-i+1if x=wi1≤i≤k-10,k-1if x=u1.For 2≤i≤k-1, the multirepresentation of ui with respect to W is(8)mrui∣W=1,i-1∪2,i∪2,k-i.For 1≤i≤k-1, the multirepresentation of vi with respect to W is(9)mrvi∣W=2,i∪3,i+1∪3,k-i+1.Therefore, W is a multiresolving set of G with W=k. Hence, dimM(G)=k.
Case 2 (n>3(k-1)). Let H be a graph obtained from the graph G in Case 1 by adding the path P=(x1,x2,…,xn-3(k-1)) and joining x1 to vk-1 and wk-1, 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 l-multiresolving set of H with 1≤l≤k-1. We claim that a set W=w1,w2,…,wk-1∪u1 is a multiresolving set of H. For vertices in V(H)-x1,x2,…,xn-3k-1, their multirepresentations with respect to W are the same as in Case 1. For 1≤i≤n-3(k-1), the multirepresentation of xi with respect to W is (10)mrxi∣W=i,i+k-1∪i+3,i+k.Hence, W is a multiresolving set of H with W=k, and so dimM(H)=k.
A connected graph G in Case 1.
A connected graph H in Case 2.
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 A={a1,a2,…,ak}∣ai∈Z for 1≤i≤k be a collection of multisets. For an integer c, we define (11)a1,a2,…,ak+c,c,…,c=a1+c,a2+c,…,ak+c,where a1,a2,…,ak∈A. Let W be a set of vertices of a connected graph G and let u and v be vertices of G. A multisimilar relation RW with respect to W on a vertex set V(G) is defined by uRWv if there is an integer cW(u,v) such that(12)mru∣W=mrv∣W+cWu,v,cWu,v,…,cWu,v.An integer cW(u,v) satisfying (12) is called a multisimilar constant of uRWv or simply a multisimilar constant. Clearly, RW is an equivalence relation on V(G). For each vertex u in V(G), let [u]W denote the multisimilar equivalence class of u with respect to W. Then(13)x∈uWif and only if mrx∣W=mru∣W+cWx,u,cWx,u,…,cWx,u,where cW(x,u) is a multisimilar constant. Observe that if x∈[u]W, then there is a multisimilar constant cW(x,u) with a property that, for every vertex w∈W, there is a corresponding vertex w′∈W such that(14)dx,w=du,w′+cWx,u.With this observation, we may as well say that x∈[u]W if and only if there are multisimilar constant cW(x,u) and a bijective function f on W defined as (15)fw=w′whenever dx,w=du,w′+cWx,u.The function f is called a multisimilar function of xRWu or a multisimilar function if there is no ambiguity. Consequently, it is not surprising that an inverse function f-1 is also multisimilar function of vRWu with a multisimilar constant cW(v,u)=-cW(u,v).
To illustrate these concepts, consider a vertex u in a connected graph G of Figure 1 and the set W=w,y,z. There is only one vertex x in V(G)-u such that x is related to u by a multisimilar relation RW with a multisimilar constant cW(x,u)=-1; that is, (16)mrx∣W=mru∣W+-1,-1,-1.Therefore, [u]W=u,x. Thus, a multisimilar function f of xRWu is defined by (17)fw=y,fy=wand fz=z.Moreover, there is another multisimilar function f′ of xRWu; that is, (18)f′w=w,f′y=yand f′z=z.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 W is not necessarily unique.
More generally, for a vertex u and a set W of vertices of a connected graph G, let mr(u∣W)=r1·a1,r2·a2,…,rl·al, where a1<a2<⋯<al and ri is a repetition number of type ai for each i with 1≤i≤l. If u∈[v]W, where v∈V(G), then it follows by (13) and (14) that, for each type of mr(u∣W), there is a corresponding type of mr(v∣W) such that their repetition numbers are equal. Therefore, we may assume that mr(v∣W)=r1·b1,r2·b2,…,rl·bl, where b1<b2<⋯<bl. For each integer i with 1≤i≤l, let Ai={w∈W∣d(u,w)=ai} and Bi={w∈W∣d(v,w)=bi}. Then the types of mr(u∣W) partition W into l sets A1,A2,…,Al. On the other hand, W is also partitioned into l sets B1,B2,…,Bl depending on the types of mr(v∣W). Hence, the multisimilar function f of uRWv has the property that, for every vertex w∈Ai, there is a vertex w′∈Bi such that (19)fw=w′,where 1≤i≤l. Indeed, there are r1!r2!⋯rl! distinct multisimilar functions of uRWv. These observations yield the following result.
Theorem 6.
Let W be a set of vertices of a connected graph G and let u and v be vertices of G such that u∈[v]W. Suppose that mr(u∣W)=r1·a1,r2·a2,…,rl·al, where a1<a2<⋯<al and ri is a repetition number of type ai for each i with 1≤i≤l. Then
mr(v∣W)=r1·b1,r2·b2,…,rl·bl for some integers b1,b2,…,bl with b1<b2<⋯<bl,
there is a multisimilar function f of uRWv such that f(wi)=wi′, where d(u,wi)=ai and d(v,wi′)=bi for each i with 1≤i≤l,
there are r1!r2!⋯rl! distinct multisimilar functions of uRWv.
By Theorem 6, the following result is obtained.
Corollary 7.
Let W be a set of vertices of a connected graph G and let u and v be vertices of G such that u∈[v]W with a multisimilar constant cW(u,v). Then
if M1 and M2 are the maximum elements of mr(u∣W) and mr(v∣W), respectively, then M1=M2+cW(u,v),
if m1 and m2 are the minimum elements of mr(u∣W) and mr(v∣W), respectively, then m1=m2+cW(u,v).
Proof.
Suppose that u∈[v]W. Let mr(u∣W)=r1·a1,r2·a2,…,rl·al and mr(v∣W)=r1·b1,r2·b2,…,rl·bl, where a1<a2<⋯<al and b1<b2<⋯<bl. Since M1 and M2 are the maximum elements of mr(u∣W) and mr(v∣W), respectively, there are vertices w and w′ in W such that M1=d(u,w)=al and M2=d(v,w′)=bl. It follows by Theorem 6 that there is a multisimilar function f of uRWv such that f(w)=w′. Then d(u,w)=d(v,w′)+cW(u,v), where cW(u,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.
Proposition 8.
Let W be a set of vertices of a connected graph G and let u and v be vertices of G such that u∈[v]W. Then mr(u∣W) and mr(v∣W) have the same minimum (or maximum) element if and only if mr(u∣W)=mr(v∣W).
Proof.
If mr(u∣W)=mr(v∣W), then the minimum (and maximum) elements of mr(u∣W) and mr(v∣W) are the same. For the converse, assume that m1 and m2 are the minimum elements of mr(u∣W) and mr(v∣W), respectively, such that m1=m2. Since u∈[v]W, there is a multisimilar constant cW(u,v) such that (20)mru∣W=mrv∣W+cWu,v,cWu,v,…,cWu,v.By Corollary 7 (ii), it follows that m1=m2+cW(u,v). Thus, cW(u,v)=0. Hence, mr(u∣W)=mr(v∣W). Similarly, if mr(u∣W) and mr(v∣W) have the same maximum element, then mr(u∣W)=mr(v∣W).
Theorem 9.
If W is a multiresolving set of a connected graph G, then the cardinality of multisimilar equivalence class of each vertex of G with respect to W is at most diam(G)+1.
Proof.
Assume, to the contrary, that there is a vertex v of G such that [v]W has the cardinality at least diam(G)+2. Since the minimum elements of multirepresentations of vertices in [v]W with respect to W have at most diam(G)+1 distinct values, there are at least two vertices x and y in [v]W having the same value of the minimum element of mr(x∣W) and mr(y∣W). It follows by Proposition 8 that mr(x∣W)=mr(y∣W), contradicting the fact that W is a multiresolving set of G.
We can show that the upper bound in Theorem 9 is sharp. Consider the path Pn=(v1,v2,…,vn). We have that diam(Pn)=n-1 and the set W=v1 is a multiresolving set of Pn. Thus, [v1]W contains all vertices of Pn, and so [v1]W=n.
In the last result, we describe the properties of a multisimilar equivalence classes with respect to a set of vertices.
Theorem 10.
Let u and v be vertices of a connected graph G and let W be a set of vertices of G. Then
if [u]W≠[v]W, then mr(x∣W)≠mr(y∣W) for all x∈[u]W and y∈[v]W,
if [u]W=u for all u∈V(G), then W is a multiresolving set of G.
Proof.
(i) Assume, to the contrary, that there exist two distinct vertices x∈[u]W and y∈[v]W such that mr(x∣W)=mr(y∣W). Then there are multisimilar constants cW(x,u) and cW(y,v) such that mr(x∣W)=mr(u∣W)+cWx,u,cWx,u,…,cWx,u and mr(y∣W)=mr(v∣W)+cWy,v,cWy,v,…,cWy,v. Therefore, (21)mru∣W+cWx,u,cWx,u,…,cWx,u=mrv∣W+cWy,v,cWy,v,…,cWy,v.Thus, mr(u∣W)=mr(v∣W)+cWy,v-cWx,u,cWy,v-cWx,u,…,cWy,v-cWx,u. Hence, u belongs to [v]W, which is a contradiction.
(ii) Assume, to the contrary, that W is not a multiresolving set of G. Then there exist two distinct vertices x and y such that mr(x∣W)=mr(y∣W). Hence, y belongs to [x]W, producing a contradiction.
4. Final Remarks
The complete graph Kn is only one graph that its dimension is n-1 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 G is a connected graph such that dimM(G) is defined, then dimM(G)≤n-2.
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.
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