On the local multiset dimension of graph with homogenous pendant edges

Let G be a connected graph with E as edge set and V as vertex set. rm (v|W) = {d(v, s 1), d(v, s 2),…, d(v, sk )} is the multiset representation of a vertex v of G with respect to W where d(v, si ) is a distance between of the vertex v and the vertices in W for k—ordered set W = {s 1, s 2,…,sk } of vertex set G. If rm (v|W) = rm (u|W) for every pair u, v of adjacent vertices of G, we called it as local resolving set of G. The minimum cardinality of local resolving set W is called local multiset dimension. It is denoted by μl (G). Hi ≅ H, for all i ∈ V(G). If H ≅ K 1, G ⊙ H is equal to the graph produced by adding one pendant edge to every vertex of G. If H ≅ mK 1 where mK 1 is union of trivial graph K 1, G ⊙ H is equal to the graph produced by adding one m pendant edge to every vertex of G. In this paper, we analyze the exact value of local multiset dimension on some graphs with homogeneous pendant edges.


Introduction
We concern on simple and undirected graphs. Metric dimension is one of the topic in graph theory which has many applications in real life. Navigation system is one of application that used metric dimension topic. By using metric dimension, we can make an effective coordinate for navigation system. Suppose, W = {s 1 , s 2 , . . . , s k } be the ordered set of vertices of a graph G. We can called W as the resolving set of G if the distinct vertices of G of the set W have distinct representations with respect to W , where the representation of v with respect to W is k-vector r(v|W ) = (d(v, s 1 ), d(v, s 2 ), ..., d(v, s k )). The metric dimension of G is the minimum cardinality of resolving set W of G is called, denoted by dim(G) [9,8].
The development of this theory has been carry out extensively. There are many types of metric dimension development, such as: partition dimension, edge metric dimension, star partion dimension, multiset dimension, etc. We can see the detail of these topics in [5,9,1]. In this paper, we develop a new study about multiset dimension. Before we go on to the development of multiset dimension, let firstly we discuss the definion of multiset dimension. Suppose W = {s 1 , s 2 , . . . , s k } is a subset of vertex set V (G). Let we define d(v, s i ) as the distance between of v and the vertices in W . r m (v|W ) = {d(v, s 1 ), d(v, s 2 ), . . . , d(v, s k )} is the representation multiset of a vertex v of G with respect to W . We say that the resolving set W as a resolving set of G if r m (v|W ) = r m (u|W ) for every pair u and v. The minimum resolving set W is a multiset basis of G. The multiset dimension is the cardinality of multiset basis of  Figure 1. A cycle graph order 6 with 3 homogenous pendant edges C 6 3K 1 with local multiset dimension 1 graph G. It is denoted by md(G). We can see the explanation of this study in [12]. Now, we develop a new study in metric dimension namely local multiset dimension. This study was the combination of multiset dimension and local metric dimension.
For more detail about definition and graph terminology, we can see [3,4,7,10]. If we consider the ordered sets W of any vertices in graph G for which any two vertices of G having the same representation with respect to W are not adjacent in G. W is called a local resolving set of G if r(u|W ) = r(v|W ) for every pair u, v of adjacent vertices of G. The local metric dimension of G is the minimum cardinality of local resolving set is. We denoted it by ldim(G) [6]. There are many studies about the local metric dimension. This topic has been studied by [6] and some related topic in the develompent of metcur dimension has been studied by [5,15,2]. In this paper, we are focused on determining the local multiset dimension of graph with homogenous pendant edges. If H ∼ = mK 1 where mK 1 is union of trivial graph K 1 , G H is equal to the graph produced by adding one m pendant edge to every vertex of G. In this paper, we search some exact value of local multiset dimension of some graph with homogeneous pendant edges.

Local Multiset Dimension
In this section, we will discuss about the definiton and example of local multiset dimension. The local multiset dimension is the development of metric dimension study. This topic was introduced by Ridho in [14]. The definion of local multiset dimension is as follows:  The minimum cardinality of local resolving set W is called as local multiset dimension. It is denoted by µ l (G).
We can see the illustration of local multiset dimension concept in 1 and the illustration of multiset dimension concept Figure 2. Based of Figure 2 and 1, we can see the differences between multiset dimension and local multiset dimension. Based on figure 2, the resolving set of thus the cardinality of multiset dimension of C 6 3K 1 is 3 or we can write md(G) = 3. Consider to the representations of v ∈ V (C 6 3K 1 ) with respect to W in figure 2, we can see that the repsesentation of each vertex is distinct. While in local multiset dimension of C 6 3K 1 , we have local recolving set v 1 or W = {v 1 }. Therefore, we get the representation of the vertices of C 6 3K 1 with respect to W are as follows: with the value of j is 1 ≤ j ≤ 3. We can see that the representations of adjacent vertices in cycle graph order 6 with homogonous pendant edges v ∈ V (C 6 3K 1 ) with respect to W are distinct. Thus, we can conclude that µ l (C 6 3K 1 ) = 1 .

Main results
Before we analyze the exact value of local multiset dimension of a graph with homogeneous pendant edges, let firstly we discuss about the properties of local multiset dimension of complete graph and tree graph. The following theorems are useful for determining the local multiset dimension of a graph with homogeneous pendant edges in the next theorem. These theorem are already discussed and proved by Ridho Et. Al in. [14]. Here is the theorem of local multiset dimension of complete graph: Theorem 3.1 Let K n be a complete graph with n ≥ 3, the local multiset dimension of K n is ∞ The next theorem is local multiset dimension of complete k−ary tree of height h: Lemma 1 Let T n be a tree graph of order n, we have µ l (T ) ≥ 1 Now, we go on to the new results about local metric dimension of graph with homogeneous pendant edge which have been discovered. Here are the results: Theorem 3.2 Let C n mK 1 be a cycle graph with homogeneous pendant edges. For n ≥ 4, the local multiset dimension of C n mK 1 is The cardinality of vertex set is n + nm and the cardinality of edge set is n + nm. The proof of this theorem is divided into two cases as follows.
Case 1: For n is even, Based on theorem 1, we have if K n be a complete graph with n ≥ 3, the local multiset dimension of K n is µ l (K n ) = ∞. We know that C n mK 1 is not a complete graph. We have known that the lower bound of local multiset dimension of cycle graph with homogeneous pendant edges for n even is 1 or we can write µ l (C n mK 1 ) ≤ 1. Next, we sholud show that the upper bound of local multiset dimension of cycle is 1 or µ l (C n mK 1 ) ≤ 1. Let W = {v 1 }, the representation of vertices v ∈ V (C n mK 1 ) respect to W as follows.
It can be seen that r m (v i |W ) = r m (v i+1 |W ) with v i and v i+1 are adjacent for 1 ≤ i ≤ n−1. It also can be seen that r m (y j i |W ) = r m (y j+1 i |W ) with y j+1 i and y j i are adjacent for 1 ≤ j ≤ n. Thus, we get the upper bound of local multiset dimension of cycle C n mK 1 is 1 or µ l (C n mK 1 ) ≤ 1. It concluded that µ l (C n mK 1 ) = 1 for n is even.
Case 2: If n is element of odd number, It will be analyzed that lower bound of the local multiset dimension of C n mK 1 is 2 or µ l (C n mK 1 ) ≥ 2. By using contradiction, let µ l (C n mK 1 ) < 2, suppose the local resolving set is 1 or W = {u}. Thus, we will have some conditions as follow: There are some adjacent vertices which have same representation. it is a contradiction. b) The next condition, let u ∈ W are in cycle graph. We know that there exist path P m , where m is even and connected with vertex u. Suppose that the vertices in P m be v 1 , v 2 . . . , v n−1 ∈ V (P m ) for n ∈ Z + . c) Based on point b, we can see that . Therefore, it is a contradict with our definion of local multiset dimension.
Based on the analysis above, it can be seen that the lower bound of local multiset dimension of C n mK 1 is 2 or µ l (C n mK 1 ) ≥ 2. The next proof is the proof of C n mK 1 local multiset dimension upper bound. We will show the upper bound of the local multiset dimension of C n mK 1 is 2 or µ l (C n mK 1 ) ≤ 2. let W = {y 1 , x 2 } be the local resolving set of C n mK 1 . The representation v respect to W in C n mK 1 are as follows: Theorem 3.3 Let P n mK 1 be a path graph with homogenous pendant edges. For n ≥ 3, the local multiset dimension of P n mK 1 is 1.
Proof. The path P n mK 1 is a tree graph. The vertex set V (P n mK 1 ) = {v 1 , v 2 , . . . , v n } ∪ {y j i } and edge set E(P n mK The cardinality of vertex set is n + m or |V (P n mK 1 )| = n + nm and the cardinality of edge set respectively is n − 1 + nm or |E(P n mK 1 )| = n − 1 + nm.
According to Lemma 1, the lower bound of local multiset dimension of tree graph T is µ l (T ) ≥ 1. We know that P n mK 1 is tree graph such that µ l (P n mK 1 ) ≥ 1. Thus, the local multiset dimension of P n mK 1 attain the lower bound. The next step, we will prove the upper bound of local multiset dimension of path P n mK 1 is 1 or µ l (P n mK 1 ) ≤ 1. Let we assume that W = {v 1 }. We got the representation of vertices v ∈ V (P n mK 1 ) respect to W are as follow:  6 We can see that r m (v i |W ) = r m (v i+1 |W ). Therefore, we attain the upper bound of local multiset dimension of P n mK 1 is µ l (P n mK 1 ) ≤ 1. Since we have proved the lower bound and upper bound of local multiset dimension of P n mK 1 , it can be concluded that µ l (P n ) = 1 Theorem 3.4 Let S n mK 1 be a star graph with homogenous pendant edges. For n ≥ 3, the local multiset dimension of S n mK 1 is 1.
Proof.The star with homogenous pendant edges S n mK 1 is a tree graph with 2n vertices.
The cardinality of vertex set is 2n + 1 or |V (S n mK 1 )| = 2n+1 and and the cardinality of edge set respectively is |E(S n mK 1 )| = 2n+1.
Based on Lemma 1, we have the lower bound of local multiset dimension of tree graph T is µ l (T ) ≥ 1. We know that S n mK 1 is tree graph such that µ l (S n mK 1 ) ≥ 1. In other word, the local multiset dimension of S n mK 1 attain the lower bound. Furthermore, we will analyze the local multiset dimension of S n mK 1 upper bound. We wil show that µ l (S n mK 1 ) ≤ 1. By assuming W = {c}, we get the representation of vertices v ∈ V (S n mK 1 ) respect to W are as follow: Hence, it is proved that the upper bound of local multiset dimension of S n mK 1 is 1 or µ l (S n mK 1 ) ≤ 1. Since we have µ l (S n mK 1 ) ≤ 1 and µ l (S n mK 1 ) ≥ 1, it can be concluded that µ l (S n mK 1 ) = 1.
Theorem 3.5 Let K n mK 1 be a complete graph with homogenous pendant edges. For n ≥ 3, the local multiset dimension of K n mK 1 is ∞ Proof : Let K n mK 1 be a complete graph with homogenous pendant edges with n+m vertices. The vertex set V (K n mK 1 ) = {v 1 , v 2 , . . . , v n } ∪ {v j i ; 1 ≤ i ≤ n − 1; 1 ≤ j ≤ m} and edge set The vertex set cardinality of K n mK 1 is n + m or |K n mK 1 | = n + m and edge set cardinality respectively is |E(K n )| = n(n−1) 2 + nm. We use contradiction in proving this theorem. K n mK 1 consist of complete graph by adding one pendant egdges in every vertex in complete graph. In order to prove this theorem, let we consider the complete graph. Suppose that all vertices in W have distance 1 and the local resolving set of complete graph K n is W . Thus, we will have the following conditions: • If we take W = {v 1 }, then r ( v 1 |W ) = {0} and r m (v 2 |W ) = r m (v 3 |W ) = · · · = r m (v n−1 |W ) = r m (v n |W ) = {1}, we know that v 2 , v 3 , . . . , v n are adjacent such that it is a contradiction. • If we take W = {v 1 , v 2 }, then r ( v 1 |W ) = r ( v 2 |W ) = {0, 1} and r m (v 3 |W ) = · · · = r m (v n−1 |W ) = r m (v n |W ) = {1 2 }, we know that , v 3 , . . . , v n are adjacent such that it is a contradiction. • If we take W = {v 1 , v 2 , . . . , v k } for 2 ≤ k ≤ n−1, then r ( v 1 |W ) = · · · = r ( v k |W ) = {0, 1 k−1 } and r m (v k+1 |W ) = · · · = r m (v n |W ) = {1 k }, we know that v k+1 , . . . , v n are adjacent such that it is a contradiction. • If W are in pendant v j i , then there will be exist some adjacent vertices v i and v i+1 in complete graph who have the same resolving set, it is a contradiction