A Note on Multiset Dimension and Local Multiset Dimension of Graphs

All graphs in this paper are nontrivial and connected simple graphs. For a set W = {s1, s2, . . . , sk} of vertices of G, the multiset representation of a vertex v of G with respect to W is r(v|W ) = {d(v, s1), d(v, s2), . . . , d(v, sk)} where d(v, si) is the distance between of v and si. If the representation r(v|W ) ̸= r(u|W ) for every pair of vertices u, v of a graph G, the W is called the resolving set of G, and the cardinality of a minimum resolving set is called the multiset dimension, denoted by md(G). A set W is a local resolving set of G if r(v|W ) ̸= r(u|W ) for every pair of adjacent vertices u, v of a graph G. The cardinality of a minimum local resolving set W is called local multiset dimension, denoted by μl(G). In our paper, we discuss the relationship between the multiset dimension and local multiset dimension of graphs and establish bounds of local multiset dimension for some families of graph.


Introduction
All graphs discussed in this paper are finite, simple and connected graph. The cartesian product graph of G 1 and G 2 , denoted by G 1 × G 2 , is the graph with vertex set V (G 1 ) × V (G 2 ) where vertex (x, u) is adjacent to vertex (y, v) whenever xy ∈ E(G 1 ) and u − v, or x − y and uv ∈ V (G 2 ). In the rest of the paper, we use the terminology defined in [1,2,3]. The application of metric dimension in networks is one of the describe navigation robots. The each place is called vertices and edges 0denote the connections0between0vertices. The minimum0number of the robots required to locate0each and the vertex of a some network is called as metric0dimension, for more detail this application in [4].
The concept of metric0dimension was independently0introduced by Slater [5] and Harary and Melter [6]. In his paper, Slater called this0concept the locating0set. Let u, v be two vertices in G, the distance d (u, v) is the length of a shortest path between two vertices u and v in graph G. An ordered0set W = {w 1 , w 2 , ..., w k } subset of vertex set V (G). The representation r (v|W ) of v with0respect to W is the ordered k-tuple r(v|W ) = (d(v, w 1 ), d(v, w 2 ), ..., d(v, w k )). The set W is called the resolving0set of G if every0vertices of G have distinct0representation with respect to W . Let u and v be any two0vertices in G if r(u|W ) = r (v|W ) implies that u = v. Hence if W is a resolving0set of cardinality k for a graph G, then the representation set r(v|W ), v ∈ V (G) consists of |V (G)| distinct k-vector. The minimum0cardinality of resolving0set of a graph G is called metric0dimension of G, denoted by dim(G). Simanjuntak et al [7] introduced the definition of multiset0dimension of G. Let G be a connected0graph with vertex set V (G). Suppose W = {s 1 , s 2 , . . . , s k } is a subset (note, not an ordered set as in metric dimension) of the vertex set V (G), the representation0multiset of a vertex v of G with0respect to W is the multiset r(v|W ) = {d(v, s 1 ), d(v, s 2 ), . . . , d(v, s k )} where d(v, s i ) is the distances between of v and the vertices in W together with their0multiplicities. A resolving set having minimum cardinality is called a multiset0basis. If G has a multiset basis, then its cardinality is called the multiset dimension of G, denoted by md(G). There are some related research about this topic in [9,10,11].
Alfarisi, et. al [8] defined a new notion based on the multiset dimension of G, namely a local0multiset dimension. The definition of local0multiset dimension is below: is a multiset of distances between of v and the vertices in W together with their0multiplicities. The resolving set W is a local resolving set of G if r(v|W ) ̸ = r(u|W ) for every pair of adjacent vertices u, v of a graph G. The minimum local0resolving set W is called local multiset dimension, denoted by µ l (G).
We illustrate this concept in Figure 1. In this case, the resolving set is W = {v 2 , v 3 , v 6 }, shown in Figure 1 (a). The multiset dimension is md(G) = 3. The representations of v ∈ V (G) with respect to W are all distinct . For the local multiset dimension, we only need to make sure the adjacent vertices having distinct representations. Thus we could have the local resolving set W = {v 1 }, shown in Figure 1 (b). Thus, the local multiset dimension is µ l (G) = 1.

Multiset Dimension
Different to the metric dimension, given a multiset basis, it is impossible to construct the original graph from the representation of the vertices. Fig 3 gives an example of two non-isomorphic graphs with the same multiset basis and representations for vertices.

Lemma 2.1
The multiset dimension is not monotonic to the number of vertices and the number of edges of a graph.
Let G be a connected graphs. The number of vertices, edges and the multiset dimension do not show a monotonic relationship. Assume the graph G with n vertices has md(G) = k, if we put m vertices in graph G, then we get new graphs G ′ with n + m vertices such that we have some condition as follows: We use a counter example for showing the Lemma 2.1. Assume the number of vertices and edges of G increase, the multiset dimension of G increase or decrease (monotonically). We choose a unicyclic graph G as example. From Figure 2 (b) we increase the vertices and edges in graph G which have the multiset dimension of G ∪ {v} namely md(G ∪ {v}) = 3 where v is a vertex not in the graph G. Furthermore, we can say that if we increase the number of vertices and edges in graphs G, then the number of resolving set does not increase or the multiset dimension is constant. it is a counter example of Lemma 2.1.
In Simanjuntak,et. al. [7], some bounds are given for the multiset dimension of graphs. For example, Theorem 2.1 [7] Let G be a graph other than a path. Then md(G) ≥ 3 If we look at the resolving set, since the vertex has distance 0 to itself, then it is easy to get a better bound than Theorem 2.2. For positive integers n and d, we define f (n, d) to be the least positive integer k for which Proof. Let W be a multiset basis of G having k vertices. If x is a vertex in W , then r(x|W ) = {0, 1 m1 , 2 m2 , . . . , d m d }, where m 1 + m 2 + · · · + m d = k − 1 and 0m i k for each i = 1, 2, . . . , k. Then there are C(k − 1 + d − 2, d − 2) different possibilities for representation of x. Since we have k vertices in W , then Furthermore, look at the degree of the graph, we could have the following bounds.

Theorem 2.3
Let G be a connected graphs and let d be the maximum degree of G and md(G) = k, we have for k ≥ 3 and d < 3k.
Proof . As md(G) = k, so let W be a multiset basis of G having k vertices. If x is a vertex not in W , then r(x|W ) = {0, 1 m1 , 2 m2 , . . . , d m d }, where m 1 + m 2 + · · · + m d = k and 0 ≤ m i ≤ k for each i = 1, 2, . . . , k. As W is a minimum resolving set, removing a vertex from W , there will be two vertices in the graph G which have the same representation. Let's assume that there are two vertices v x and v y which are in G. Formally, we have Considering the neighbour of these two vertices, if v x has distance t to a vertex w ∈ W , then the neighbour of v x would have distance t − 1 or t or t + 1 to w. So, if v x has distance 1 to m 1 vertices in W , then among v x 's neighbour, there are at most m 1 vertices having distance 0 to vertices in W , i.e. in W . if v x has distance 2 to m 2 vertices in W , then among v x 's neighbour, there are at most m 2 vertices having distance 1 to vertices in W , furthermore, v x 's neighbour having distance less than 3 to at least m 2 vertices in W . More general, if v x having distance x to m x vertices in W , then among the neighbours of v x , there are less than m x vertices of distance x − 1 to W and there are no more than m x vertices having distance x + 1 to vertices in W .
Thus, the number of different representations for the neighbour of v x is m 1 (m 1 + m 2 )(m 2 + m + 3)..(m d−1 + md)d and the representations are shared by neighbours of v x and v y . This will give us a bound for the degree, diameter and multiset dimension. 2 We shall define a new graph which is based on the well-known Hypercubes. The Hypercube is defined the graph formed from the vertices and edges of an n-dimensional hypercube, we shall remove some edges from the hypercube, denoted by AHQ n , is called almost hypercube graphs. Almost hypercube graph satisfies AHQ n = (HQ n−1 × P 2 ) − {e} for n ≥ 3, where e is correspondence edge of subgraph (HQ n−1 ) 1 and (HQ n−1 ) 2 . We know that HQ n−1 × P 2 has two isomorphic graphs (HQ n−1 ) 1 and (HQ n−1 ) 2 with {e} is the correspondence edge set.
i. If three vertices in HQ ′ 2 , then there is at least two vertices which have same representation. ii. If two vertices in HQ ′ 2 and one vertex in HQ 2 ", then always two vertices with respect to v ∈ W (HQ 2 ") which have same representation.
Based on cases above, we know that md(AHQ 3 ) ≥ 3 Case 2: We have some condition for proof this cases which divided into some cases as follows.
i. Shortest path between two vertices in the same components is within the component ii.
then there is at least two vertices which have same representation.
Based on both cases, we obtain the bounds of multiset dimension of almost hypercube graph namely md(AHQ n ) ≥ 2 n−1 − 1.

Local Multiset Dimension
In this section, we give some results about local multiset dimension of graphs. Firstly, we show that the local multiset dimension for AHQ is 1, which is different to the results we got for multiset dimension.

Corollary 3.1
The difference between Multiset dimension and local multiset dimension can be arbitrarily large.

Observation 3.1
If G is complete graph, then the graph G does not have a local resolving set.

Lemma 3.1
The local multiset dimension is not monotonic to the number of vertices and the number of edges of a graph.
Let G be a connected graphs. The number of vertices, edges and the local multiset dimension do not show a monotonic relationship. Assume the graph G with n vertices has µ l (G) = k, if we put m vertices in graph G, then we get new graphs G ′ with n + m vertices such that we have some condition as follows: We use a counter example for showing the Lemma 3.1. Assume the number of vertices and edges of G increase, the local multiset dimension of G increase or decrease (monotonically). We choose a bipartite graph B 5,5 as example. From Figure 4 (a) the local multiset dimension of B 5,5 is µ l (B 5,5 ) = 1, we add some edges in B 5,5 in Figure 4 (b) such that we have the local multiset dimension of B 5,5 ∪ {e} namely µ l (B 5,5 ∪ {e}) = 1. Figure  4 (c) we increase the vertices and edges in graph B 5,5 which have the local multiset dimension of B 5,5 ∪ {v} namely µ l (B 5,5 ∪ {v}) = 1 where v is a vertex not in the graph B 5,5 . Furthermore, we can say that if we increase the number of vertices and edges in graphs B 5,5 , then the number of resolving set does not increase or the local multiset dimension is constant. it is a counter example of Lemma 3.1.
The following, we show the new bound of local multiset dimension of cartesian product of graphs. Let G 1 and G 2 be two connected graphs.

Lemma 3.2
Given that two connected graphs G 1 and G 2 , µ l (G 1 × G 2 ) ≥ min{µ l (G 1 ), µ l (G 2 )} Proof. A graph G 1 has n 1 vertices and G 2 has n 2 vertices. The cartesian product graph of G 1 and G 2 , denoted by G 1 × G 2 , is the graph with vertex set V (G 1 ) × V (G 2 ) where vertex (x, u) is adjacent to vertex (y, v) whenever xy ∈ E(G 1 ) and u − v, or x − y and uv ∈ V (G 2 ). For a fixed x of G 1 , the vertices {(x, u)|u ∈ V (G 2 )} induces a subgraph of G 1 × G 2 isomorphic to G 2 and we call it as G 2 -layer. Such that, we have G 1 -layers or G 2 -layers. Assume that we have local multiset dimension of G 1 and G 2 , respectively are µ l (G 1 ) = k 1 and µ l (G 2 ) = k 2 .
Based on both cases, we can claim that |W The cartesian product of graph G and tree graph T with characterization for µ l (G) = 1 and we get the results as follows.

Theorem 3.1
Given that a connected graph G and a path P n , µ l (G × P n ) = µ l (G) Proof: The graph G × P n has n copies subgraph G i , 1 ≤ i ≤ n. Let W be a local resolving set of G = G i so that every vertices u, v ∈ V (G) for u adjacent to v has different representation. If we assume that W is a set of G × P n , then we prove that W is local resolving set of G × P n , i. We know that for every vertices u ∈ W belong to in subgraph G 1 or first copy (first layer). ii. Every two adjacent vertices u, v ∈ V (G 1 ) − W , has different representation. Since, a set W is the local resolving set of G = G 1 .

iii. For every two adjacent vertices
Based on five cases (i) − (v), W is a local resolving set of G × P n . Thus, we have upper bound of local multiset dimension of G × P n is µ l (G × P n ) ≤ µ l (G).
Furthermore, we show that the lower bound of local multiset dimension of G × P n is µ l (G × P n ) ≥ µ l (G). Assume that |W G×Pn | < |W G |, by taking |W G×Pn | = |W G | − 1.
i. For every vertices v ∈ W G×Pn belong to in subgraph G 1 such that there exists at least two adjacent vertices has same representation. ii. Let u, v ∈ V (G 1 ) where u adjacent to v, d(u, w) = d(v, w). Thus, r(u|W ) = r(v|W ). It is a contradiction. iii. If some vertices of resolving set not all in subgraph G 1 , then there is at least one vertex of W in G i , 1 ≤ i ≤ n.

iv. For any two adjacent vertices
It is a contradiction.
Based cases above, we have the local resolving set of G × P n at least |W G | or |W G×Pn | ≥ |W G |. Hence, we have lower bound of local multiset dimension of G × P n is µ l (G × P n ) ≥ µ l (G). Thus, the local multiset dimension of G × P n is µ l (G × P n ) = µ l (G). 2

Theorem 3.2
Given that a connected graph G and a tree T , µ l (G × T ) ≤ µ l (G) Proof: The graph G × T has n copies subgraph G i , 1 ≤ i ≤ n. Let W be a local resolving set of G = G i so that every vertices u, v ∈ V (G) for u adjacent to v has different representation. If we assume that W is a set of G × T , then we prove that W is local resolving set of G × T , i. We know that for every vertices u ∈ W belong to in subgraph G 1 or first copy (first layer). ii. Every two adjacent vertices u, v ∈ V (G 1 ) − W , has different representation. Since, a set W is the local Based on four cases (i) − (iv), W is a local resolving set of G × T . Thus, we have upper bound of local multiset
Thus, we obtain µ l (G × T ) ≤ 1. Thus, µ l (G × T ) = 1, for µ l (G) = 1 and any tree T . Proof: If local multiset dimension µ l (G) ̸ = 1, then we have local resolving set |W | ≥ 2. From Lemma 3.2, it states that µ l (G × T ) ≥ min{µ l (G), 1} ≥ 1. Assume that |W | = 1. There is at least two adjacent vertices which have same representation. Choose the local resolving set in G 1 . Every adjacent vertices in ( Thus, the cardinality of the local resolving set of G × T is |W | ̸ = 1, and the local multiset dimension of µ l (G × T ) ≥ 2. Proof: Alfarisi, et. al. [8] determined the local multiset dimension of C m with m is odd is 1. If the local multiset dimension of G is 1, then every two adjacent vertices has distinct representation. From Lemma 3.2 that µ l (G × C m ) ≥ min{µ l (G), µ l (C m )} ≥ min{1, 1} ≥ 1. Furthermore, we can show µ l (G × C m ) ≤ 1 as follows.
Thus, we obtain that µ l (G × C m ) ≤ 1. It concludes that µ l (G × C m ) = 1, for µ l (G) = 1 and m is even. Proof: Based on Lemma 3.1 that µ l (G 1 × G 2 ) = min{µ l (G 1 ), µ l (G 2 )}. If one of both graph has local multiset dimension at least one, then µ l (G × C m ) ≥ 1. We try construct of the sharpest lower bound of G × C m for m is odd or (µ l (G) ̸ = 1 and m is even) as follows. Case 1: For µ l (G) = 1 and m is odd We know that µ l (C m ) = 3 for n is odd and µ l (G) = 1, based Lemma 3.2 that µ l (G × C m ) ≥ min{1, 3} ≥ 1. Assume that |W | = 1, there is at least two adjacent vertices which have same representation. Choose the local resolving set in (C m ) 1 , then every adjacent vertices in ( Thus, the cardinality of the local resolving set of G × C m is |W | ̸ = 1, such that the local multiset dimension of µ l (G × C m ) ≥ 2. This illustration can be seen in Figure 8 Thus, the cardinality of the local resolving Next, we study a Hypercube graph, denoted by HQ n . Hypercube graph is the graph formed from the vertices and edges of an n-dimensional hypercube. It is the n-fold Cartesian product of the two-vertex complete graph, and decomposed into two copies of HQ n−1 connected to each other by a perfect matching. Proof: Hypercube graph satisfies HQ n = HQ n−1 × P 2 for n ≥ 0. For n = 0, we have HQ 0 isomorphic to K 1 or trivial graphs. The local multiset dimension of K 1 is µ l (HQ 0 ) = 1. To prove this theorem, we can use a mathematical induction or recursive technique below.
Thus, the local multiset dimension of hypercube HQ n is µ l (HQ n ) = 1, for n ∈ N ∪ {0}. Figure 9 is an illustration of local multiset dimension of hypercube graphs for HQ 4 .  Proof: Almost hypercube graph satisfies AHQ n = (HQ n−1 × P 2 ) − {e} for n ≥ 3, where e is correspondence edge of subgraph (HQ n−1 ) 1 and (HQ n−1 ) 2 . We know that HQ n−1 × P 2 has two isomorphic graphs (HQ n−1 ) 1 and (HQ n−1 ) 2 with {e} is the correspondence edge set. Based on Theorem 3.4 that µ l (HQ n−1 ) = 1 such that for two adjacent vertices u, v ∈ V ((HQ n−1 ) 1 ) has the distinct representation namely r(u|W ) ̸ = r(v|W ) for W in the first copy of HQ n−1 .

Corollary 3.3
For n ≥ 4, µ l (U K(C n )) = { 1, if n is even 3, if n is odd Furthermore, we characterization of relationship between multiset dimension and local multiset dimension of graphs.

Remark 3.1
The relationship of multiset dimension and local multiset dimension of graphs G, gap(md(G), µ l (G) = ∞.

Conclusion
In this paper we have given an result the lower bound of multiset dimension and local multiset dimension of graphs. Hence the following problem aries naturally.

Open Problem 4.1
Determine the local multiset dimension of family graph namely family tree, unicyclic, regular graphs, and others.

Open Problem 4.2
Determine the local multiset dimension of operation graph namely corona product, joint, comb product, and others.