On the Dominant Local Resolving Set of Vertex Amalgamation Graphs


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(4) (2023), Pages 597-607 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: December 19, 2022 Reviewed: December 20, 2022 Accepted: March 16, 2023 
DOI: http://dx.doi.org/10.18860/ca.v7i4.18891  

On the Dominant Local Resolving Set of Vertex Amalgamation Graphs 

Reni Umilasari1,2, Liliek Susilowati1,*, Slamin3, Savari Prabhu4 

1,2
Department of Mathematics, Airlangga University, Surabaya 60115, Indonesia 

2Department of Informatics Engineering, University of Muhammadiyah Jember, Jember 
69121 Indonesia 

3Study Program of Informatics, University of Jember, Jember 68121 Indonesia 

4Mathematics Department, Rajalakshmi Engineering College, Chennai 602105, Tamil 
Nadu, India 

 

Email: liliek-s@fst.unair.ac.id  

ABSTRACT  

In graph theory, there is a new topic of the dominant local metric dimension which be symbolized by 
𝐷𝑑𝑖𝑚𝑙 (𝐻) and it was a combination of local metric dimension and dominating set. There are some terms in 
this topic that is dominant local resolving set and dominant local basis. An ordered subset 𝑊𝑙  is said a 
dominant local resolving set of 𝐺 if 𝑊𝑙  is dominating set and also local resolving set of 𝐺. While dominant 
local basis is a dominant local resolving set with minimum cardinality. This study uses literature study 
method by observing the local metric dimension and dominating number before detecting the dominant 
local metric dimension of the graphs. After obtaining some new results, the purpose of this research is how 
the dominant local metric dimension of vertex amalgamation product graphs. Some special graphs that be 
used are star, friendship, complete graph and complete bipartite graph. Based on all observation results, it 
can be said that the dominant local metric dimension for any vertex amalgamation product graph depends 
on the dominant local metric dimension of the copied graphs and how the terminal vertex is constructed. 
 
Copyright © 2023 by Authors, Published by CAUCHY Group. This is an open access article under the CC BY-
SA License (https://creativecommons.org/licenses/by-sa/4.0/) 
 
Keywords: dominant local metric dimension; vertex amalgamation; star; friendship; complete bipartite 

INTRODUCTION 

Metric dimension and dominating set are graph topics with numerous variations. 
For metric dimensions, there are more than five development concepts, such as partition 
dimension, local metric dimension, complement metric dimension, central metric 
dimension, fractional metric dimension, and star metric dimension. More results about 
metric dimension and its expansion can be seen at [1] about the simultaneous local metric 
dimension, the local metric dimension of amalgamation [2] and corona product of star 
and path graph [3] , complement metric dimension [4], fractional metric dimension [5], 
and the new one is star metrc dimension [6]. 

Let 𝐺 be a connected graph with vertex set 𝑉(𝐺) and edge set 𝐸(𝐺). The distance 
between any two 𝑎 and 𝑏 of 𝑉(𝐺) is denoted by 𝑑(𝑎, 𝑏). It be defined as shortest path from 
𝑎 to 𝑏. The resolving set of 𝐺 is an ordered set which can be written as 𝑊, where 𝑊 = {𝑤1,
𝑤2, … , 𝑤𝑘 } ⊆ 𝑉(𝐺) and  𝑟(𝑣|𝑊) = (𝑑(𝑣, 𝑤1), 𝑑(𝑣, 𝑤2), … . , 𝑑(𝑣, 𝑤𝑘 )) is defined as 

http://dx.doi.org/10.18860/ca.v7i4.18891
mailto:liliek-s@fst.unair.ac.id
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On the Dominant Local Resolving Set of Vertex Amalgamation Graphs 

Reni Umilasari 598 

representation of 𝑣 ⊆ 𝑉(𝐺) to 𝑊 by using the concept of distance. The rule to select 
resolving set of 𝐺 is every vertex of 𝑉(𝐺) should have different representation to 𝑊. The 
minimum cardinality of 𝑊 is called the basis of 𝐺 [7]. The number of basis is referred to 
the metric dimension because the concept of this topic is based on distance. Next, we 
introduce the differences between metric dimension and local metric dimension. In the 
metric dimension, all vertices must have different representations to the resolving set, 
whereas in the local metric dimension, only any two adjacent vertices must be different. 
It also can be said that a vertex's representation can be the same as another vertices even 
though they are not adjacent. [8]. Some examples of the local metric dimension have been 
published at [9], [10] and [11].  More clearly, there is a paper which describe the similarity 
between metric dimension and the local metric dimension [12]. In 2021, Umilasari et al 
introduced the new concept, which is a combination of dominating set and local metric 
dimension. They defined that an ordered subset 𝑊𝑙  is said a dominant local resolving set 
of 𝐺 if 𝑊𝑙  is dominating set and also local resolving set of 𝐺. For a clearer understanding 
of this term, you can see the illustration in Figure 1. 

All the vertices in the graph of Figure 1 (a) can be dominated by 𝑥4. But the vertices 
which adjacent {𝑉(𝐺)\𝑥4} have same representation to 𝑥4. While {𝑥2, 𝑥3} in Figure 1 (b) 
is a local resolving set. As can be seen, each pair of adjacent vertices has a different 
representation to {𝑥2, 𝑥3}. The problem is 𝑥5 and 𝑥6 cannot be dominated by 𝑥2 or 𝑥3.  If 
we take two vertices like in Figure 1(c), {𝑥2, 𝑥4} can dominate all vertices of the graph, the 
representation of any two vertices to {𝑥2, 𝑥4} is different. Therefore, {𝑥2, 𝑥4} are elements 
of a dominant local metric dimension of the graph. 

 

 

Figure 1. The Illustration of Dominant Local Metric Dimension of Graphs 
 

After obtaining some new results, in this paper we continue to expand on how the 
dominant local metric dimension of a vertex amalgamation product graphs. The vertex 
amalgamation product of a graph 𝐻, denoted by 𝑎𝑚𝑎𝑙(𝐻, 𝑣, 𝑘), is defined as creating a 
new graph by gluing 𝑘-copies of 𝐻 in a terminal vertex 𝑣 [13]. In this paper, we determine 
the dominant local metric dimension of the vertex amalgamation for some special graphs, 
which are star, complete graph, complete bipartite graph, and friendship graph. To make 
it easier to present each of the theorems produced, several theorems are given below, 
which can be seen in [14]. 

 
Lemma 1. Let 𝐺 be a connected graph. If there is no local dominant resolving set with 
cardinality 𝑝, then for every 𝑆 ⊆ 𝑉(𝐺) with |𝑆| < 𝑝 is not a local dominant resolving set. 
Lemma 2. Let 𝐺 be a connected graph and 𝑊𝑙 ⊆ 𝑉(𝐺).  For every 𝑣𝑖 , 𝑣𝑗 ∈ 𝑊𝑙  then 

𝑟(𝑣𝑖 |𝑊𝑙 ) ≠ 𝑟(𝑣𝑗 |𝑊𝑙 ).  

Some new results about the dominant local metric dimension of star, complete 



On the Dominant Local Resolving Set of Vertex Amalgamation Graphs 

Reni Umilasari 599 

graph, complete bipartite graph, and friendship graph are given in Table 1.  
 
  



On the Dominant Local Resolving Set of Vertex Amalgamation Graphs 

Reni Umilasari 600 

Table 1. Dominant Local Metric Dimension of Special Graphs [14][15][16] 

Graphs Dominating 
Number 
(𝜸(𝑮)) 

Local Metric 
Dimension 

(𝐝𝐢𝐦𝒍(𝑮)) 

Dominan Local 
Metric Dimension 

(𝐝𝐢𝐦𝒍(𝑮)) 

Star (𝑆𝑛) 𝛾 (𝑆𝑛) = 1 dim𝑙  (𝑆𝑛) = 1 Ddim𝑙 (𝑆𝑛) = 1 

Complete (𝐾𝑛) 𝛾 (𝐾𝑛) = 1 dim𝑙  (𝐾𝑛) = 𝑛 − 1 Ddim𝑙 (𝐾𝑛) = 𝑛 − 1 

Complete Bipartite 
(𝐾𝑚,𝑛) 

𝛾(𝐾𝑚,𝑛) = 2 dim𝑙 (𝐾𝑚,𝑛) = 2 
Ddim𝑙 (𝐾𝑚,𝑛) = 2 

Friendship (𝐹𝑛 ) 𝛾(𝐹𝑛 ) = 1 dim𝑙 (𝐹𝑛) = 𝑛 Ddim𝑙 (𝐹𝑛) = 𝑛 

 

METHODS 

In this research, there are several procedures. We start by determining the special 
graphs to be operated by the vertex amalgamation product and observing the local metric 
dimensions and dominating number of the graphs. Then, we construct the vertex 
amalgamation product graphs from the special graphs that we have chosen. We continue 
by labeling the vertex and attempting to find the least dominant local basis. This is 
accomplished by observing and recording the representation of each vertex which can be 
dominated and has different representation from the local resolving set (two non-
neighbouring vertex can have the same representation). The minimum local dominant 
basis is then determined. In summary, the procedures of the research can be seen in the 
following flowchart in Figure 2. We also give some examples of each step below. 
a) Let 𝐺 = 𝑃4 
b) 𝑑𝑖𝑚𝑙 (𝑃4) = 1 and 𝛾(𝑃4) = 2 , it can be seen at [16] 
c) Let |𝑊𝑙 | = 1, 𝑊𝑙 = {𝑣1} 

 
Illustration: 
 

 
 
Based on the illustration above, 𝑣1 can’t dominate 𝑣3 and 𝑣4. When we choose 𝑊𝑙 =
{𝑣2}, 𝑊𝑙 = {𝑣3}, 𝑊𝑙 = {𝑣4} the condition remains the same. Minimally, there exist one 
vertex that can’t be dominated. 

d) Let |𝑊𝑙 | = 2, 𝑊𝑙 = {𝑣1, 𝑣2} 
 
Illustration: 
 

 
 
We can see that 𝑣4 can’t be dominated by 𝑣1 or 𝑣2 .  

e) Let |𝑊𝑙 | = 2, 𝑊𝑙 = {𝑣2, 𝑣3} 
 
Illustration: 
 

 
Since ∀𝑣𝑖 𝑣𝑗 ∈ 𝐸(𝑃4), 𝑟(𝑣𝑖 |𝑊𝑙 ) ≠ 𝑟(𝑣𝑗 |𝑊𝑙 ) then 𝑊𝑙  is basis local of 𝑃4. All vertices of 

𝑉(𝑃4) can be dominated by 𝑣2 and 𝑣3. Therefore, 𝑊𝑙  is dominant local basis of 𝑃4 or 
𝐷𝑑𝑖𝑚𝑙 (𝑃4) = 2. 

To more clearly understand this research method, we can see the flowchart in Figure 2. 



On the Dominant Local Resolving Set of Vertex Amalgamation Graphs 

Reni Umilasari 601 

 

Figure 2. Flowchart for Determining the Minimum Dominant Local Resolving Set of 
Graphs 

RESULTS AND DISCUSSION  

In this section, we determine the dominant local metric dimension of the vertex 
amalgamation product for some special graphs, which are star, complete graph, complete 
bipartite graph, and friendship graph.  
Theorem 1. Let 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘) is a vertex amalgamation of star with the order of star is 
𝑛 ≥ 3, then 

𝐷𝑑𝑖𝑚𝑙  (𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) = {
1,   𝑣 𝑖𝑠 𝑐𝑒𝑛𝑡𝑒𝑟 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 𝑆𝑛

𝑘,   𝑣 𝑖𝑠 𝑝𝑒𝑛𝑑𝑎𝑛𝑡 𝑜𝑓 𝑆𝑛
 

  



On the Dominant Local Resolving Set of Vertex Amalgamation Graphs 

Reni Umilasari 602 

Proof.  
Case 1. 𝑣 is center vertex of star 
It is very clearly to see that 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘) ≅ 𝑆𝑛, then by the Table 1 we can conclude that 

𝐷𝑑𝑖𝑚𝑙  (𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) = 1.  ∎ 

Case 2. 𝑣 is pendant vertex of star 
Let the vertex set of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘) is 𝑉(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) =

{𝑣, 𝑣𝑗 , 𝑢1𝑗 , 𝑢2𝑗 , … , 𝑢𝑖𝑗 |𝑣, 𝑢𝑖 ∈ 𝑉(𝑆𝑛), 𝑖 = 1,2, … , 𝑛 − 2, 𝑗 = 1,2, … , 𝑘} and the edge set is 

𝐸(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) = {𝑣𝑣𝑗 , 𝑣𝑗 𝑢𝑖𝑗 |𝑖 = 1,2,3, … , 𝑛 − 2, 𝑗 = 1,2,3, … , 𝑘}. Choose 𝑊𝑙 = {𝑣𝑗 } is 

the local basis of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)  for every 𝑗 = 1,2,3, … , 𝑘, |𝑊𝑙 | = 𝑘.  We can show below 
that the representation of every two adjacent vertices of 𝑉(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) is different. 
i. For 𝑢𝑖𝑗 𝑣𝑗 ∈ 𝐸(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘))  

Since 𝑣𝑗  is element of 𝑊𝑙 , then there exist 0 on 𝑖
th element in  𝑟(𝑣𝑗 |𝑊𝑙 ), while  for 

𝑟(𝑢𝑖𝑗 |𝑊𝑙 ) there are no zero elements, hence 𝑟(𝑣𝑗 |𝑊𝑙 ) ≠ 𝑟(𝑢𝑖𝑗 |𝑊𝑙 ). 

ii. For 𝑣𝑣𝑗 ∈ 𝐸(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘))  

Since 𝑣𝑗  is element of 𝑊𝑙 , then there exist 0 on 𝑖
th element in  𝑟(𝑣𝑗 |𝑊𝑙 ), while  for 𝑟(𝑣|𝑊𝑙 ) 

there are no zero elements, hence 𝑟(𝑣𝑗 |𝑊𝑙 ) ≠ 𝑟(𝑣|𝑊𝑙 ). 

By i and ii therefore 𝑊𝑙  is a local resolving set of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘). Further, because 𝑣𝑗  is 

adjacent to 𝑣 and 𝑢𝑖𝑗 , so we can said that 𝑊𝑙  is a dominant local resolving set of 

𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘). Next, take any 𝑆 ⊆ 𝑉(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) with |𝑆| < |𝑊𝑙 |. Without loss of 
generality, let  |𝑆| = |𝑊𝑙 | − 1  with 𝑊𝑙 = {𝑣𝑗 |𝑗 = 1,2,3, … , 𝑘 − 1}, so 𝑢𝑖𝑘  are not adjacent to 

𝑆. So 𝑆 is not a dominant local resolving set of  𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘). Based on Lemma 1, any set 
𝑇 with |𝑇| < |𝑆| is not a dominant local resolving set of 𝐺. Therefore, 𝑊𝑙 = {𝑣𝑗 }  is a 

dominant local basis of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘). Then its is proven that 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) = 𝑘 

for 𝑣 is pendant vertex of star of 𝑆𝑛.  ∎ 

 

Figure 3. 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝑆4, 𝑣, 3)) = 1. 

 

 

Figure 4. 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝑆6, 𝑣, 5)) = 5 



On the Dominant Local Resolving Set of Vertex Amalgamation Graphs 

Reni Umilasari 603 

Figure 3 gives the illustration of dominant local metric dimension of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘) for 𝑣 
is the center vertex of 𝑆𝑛. While in Figure 4, 𝑣 is the pendant of Star. The next theorem, we 
will show the dominan local metric dimension of complete graph. Because the graphs are 
regular, then we can select any vertex of complete graph as the linkage vertex. 
 
Theorem 2. Let 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝) is a vertex amalgamation of complete graph with the order 
of complete graph is 𝑛 ≥ 3, then 𝐷𝑑𝑖𝑚𝑙  (𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝)) = 𝑝 × (𝐷𝑑𝑖𝑚𝑙 (𝐾𝑛) − 1). 

 
Proof. Let 𝑉(𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝)) = {𝑣, 𝑣𝑖𝑗 |𝑖 = 1,2,3, … , 𝑛 − 1, 𝑗 = 1,2,3, … , 𝑝} and the edge set 

of 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝) is 𝐸(𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝)) =

{𝑣𝑣𝑖𝑗 , 𝑣𝑥𝑗 𝑣𝑦𝑗 |𝑖 = 1,2,3, … , 𝑛 − 1, 𝑗 = 1,2,3, … , 𝑝, 𝑣𝑥 𝑣𝑦 ∈ 𝐸(𝐾𝑛), 𝑥 ≠ 𝑦 }. The 𝑗-th copy of  𝐾𝑛 

with 𝑗 = 1,2,3, … , 𝑝 is called (𝐾𝑛)𝑗 . Let 𝐵 be a local dominant basis of 𝐾𝑛, 𝐵𝑗  is a local 

dominant basis of (𝐾𝑛)𝑗 , so that for every 𝑗 = 1,2,3, … , 𝑝, |𝐵𝑖 | = |𝐵|. Select 𝑊𝑙 = ⋃ 𝐵𝑗
𝑚
𝑗=1 , 

with 𝐵𝑗 = {𝑣𝑖𝑗 |𝑗 = 1,2,3, … , 𝑛 − 2} for every 𝑗 = 1,2,3, … , 𝑝, then |𝑊𝑙 | = 𝑝(𝑛 − 2).  By 

Lemma 2 for every 𝑣𝑖𝑗 , 𝑣𝑘𝑙 ∈ 𝐵𝑖  then 𝑟(𝑣𝑖𝑗 |𝑊𝑙 ) ≠ 𝑟(𝑣𝑘𝑙 |𝑊𝑙 ). Next, we take any two 

adjacent vertices in 𝑉(𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝))\𝑊𝑙 . Let 𝑥, 𝑦 ∈ 𝑉(𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝))\𝑊𝑙 , then for 𝑥𝑦 =

𝑣, 𝑣𝑛𝑗 ∈ 𝑉(𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝))\𝑊𝑙 with 𝑗 = 1,2,3, … , 𝑝. Since 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝) is a connected 

graph, 𝑑(𝑣𝑛𝑗 , 𝑧) = 𝑑(𝑣𝑛𝑗 , 𝑣) + 𝑑(𝑣, 𝑧)for every 𝑧 ∈ 𝐵𝑖  so that  𝑑(𝑧, 𝑣𝑛𝑗 ) ≠ 𝑑(𝑧, 𝑣) caused 

𝑟(𝑣𝑛𝑗 |𝐵𝑖) ≠ 𝑟(𝑣|𝐵𝑖). Because of 𝐵𝑖 ⊆ 𝑊𝑙  then  𝑟(𝑣𝑛𝑗 |𝑊𝑙 ) ≠ 𝑟(𝑧|𝑊𝑙 ). 

Based on the explanation above,  𝑊𝑙 = ⋃ 𝐵𝑖
𝑚
𝑖=1  is a local resolving set of 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝). 

Since, every 𝑣𝑖𝑗 ∈ 𝑊𝑙  with 𝑖 = 1,2,3, … , 𝑛 − 2 and 𝑗 = 1,2,3, … , 𝑝  is adjacent to 𝑣 and 𝑣𝑛𝑗 , 

then 𝑊𝑙  is a dominating set. So that,  𝑊𝑙 = ⋃ 𝐵𝑗
𝑝
𝑗=1  is a local dominant resolving set of  

𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝). Take any 𝑆 ⊆ 𝑉(𝐺) with |𝑆| < |𝑊𝑙 |. Let |𝑆| = |𝑊𝑙 | − 1, then there exists 𝑗 
such as 𝑆 contains maximal |𝐵𝑗 | − 1 elements of (𝐾𝑛)𝑗 . Since 𝐵𝑗  is a local dominant basis 

of (𝐾𝑛)𝑗  then there exist two vertices in (𝐾𝑛)𝑗  that have the same representation toward 

𝑆, so that 𝑆 is not a local dominant resolving set of 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝). Based on Lemma 1 then 
𝑊𝑙 = ⋃ 𝐵𝑗

𝑝
𝑗=1  is a local dominant basis of 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝). By Table 1 we know that |𝐵𝑖 | =

𝐷𝑑𝑖𝑚𝑙 ((𝐾𝑛)𝑖 ) − 1, then it has been proven that 𝐷𝑑𝑖𝑚𝑙  (𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝)) = 𝑝 ×

(𝐷𝑑𝑖𝑚𝑙 (𝐾𝑛) − 1). ∎ 
The example of dominant local metric dimension of vertex amalgamation complete 

graph be given in Figure 5.  The graph show that 𝑎𝑚𝑎𝑙(𝐾4, 𝑣, 3) has the dominant local 
metric dimension equals six. 
 

 

Figure 5. 𝐷𝑑𝑖𝑚𝑙  (𝑎𝑚𝑎𝑙(𝐾4, 𝑣, 3)) = 6. 

 

Theorem 3. Let 𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝) is a vertex amalgamation of complete bipartite graph 

with the order is 𝑚, 𝑛 ≥ 2, then 𝐷𝑑𝑖𝑚𝑙  (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) = 𝑝 + 1. 



On the Dominant Local Resolving Set of Vertex Amalgamation Graphs 

Reni Umilasari 604 

Proof. Let the vertex set of 𝐾𝑚,𝑛  is 𝑉(𝐾𝑚,𝑛) = {𝑎𝑖 |𝑖 = 1,2, … , 𝑚} ∪ {𝑏𝑗 |𝑗 = 1,2, … , 𝑛}, and 

the edge set is 𝐸(𝐾𝑚,𝑛) = {𝑎𝑖 𝑏𝑗 |𝑖 = 1,2, … , 𝑚; 𝑗 = 1,2, … , 𝑛}. 𝑉 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) =

{𝑣, 𝑎𝑖𝑘 , 𝑏𝑗𝑘 |𝑖 = 2,3, … , 𝑚, 𝑗 = 1,2, … , 𝑛, 𝑘 = 1,2, … , 𝑝} and the edge set is 

𝐸 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) = {𝑣𝑏𝑗𝑘 , 𝑎𝑖𝑘 𝑏𝑗𝑘 |𝑖 = 2,3, … , 𝑚, 𝑗 = 1,2, … , 𝑛, 𝑘 = 1,2, … , 𝑝}. Choose, 

𝑊𝑙 = {𝑣, 𝑏1𝑘 } for every 𝑘 = 1,2,3, … , 𝑝, then |𝑊𝑙 | = 𝑝 + 1. We can show below that the 

representation of every two adjacent vertices of𝑉 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) is different. 

i. For 𝑣𝑏𝑗𝑘 ∈ 𝐸 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝))  

Since 𝑣 is element of 𝑊𝑙 , then there exist 0 on 1
𝑠𝑡  element in 𝑟(𝑣|𝑊𝑙 ), while  for 

𝑟(𝑏𝑗𝑘 |𝑊𝑙 ) there are no zero elements except 𝑏1𝑘  the representation to 𝑊𝑙  is 𝑟(𝑏1𝑘 |𝑊𝑙 ) =

(1,0), hence 𝑟(𝑣|𝑊𝑙 ) ≠ 𝑟(𝑏𝑗𝑘 |𝑊𝑙 ). 

ii. For 𝑎𝑖𝑘 𝑏𝑗𝑘 ∈ 𝐸 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) 

Since for 𝑖 = 2,3, … , 𝑚, 𝑗 = 1,2, … , 𝑛, 𝑑(𝑎𝑖𝑘 , 𝑣) = 𝑑(𝑎𝑖𝑘 , 𝑏𝑗𝑘 ) + 𝑑(𝑏𝑗𝑘 , 𝑣) hence 

𝑟(𝑎𝑖𝑘|𝑊𝑙 ) ≠ 𝑟(𝑏𝑗𝑘 |𝑊𝑙 ). 

From the two explanations above we know that 𝑊𝑙  is the local resolving set of 
𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝). Since 𝑣 is adjacent to 𝑏𝑗𝑘  for 𝑗 = 1,2, … , 𝑛 and 𝑘 = 1,2, … , 𝑝. The vertex 

𝑏1𝑘  is adjacent to 𝑎𝑖𝑘  for 𝑖 = 2,3, … , 𝑚 and 𝑘 = 1,2, … , 𝑝, thus 𝑊𝑙  is dominant local 

resolving set of 𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝). Take any 𝑆 ⊆ 𝑉(𝐺) with |𝑆| < |𝑊𝑙 |. Let |𝑆| = |𝑊𝑙 | − 1 

the two possibilities below: 
a. If 𝑣 ∉ 𝑊𝑙  

𝑣 ∉ 𝑊𝑙 , then all vertices 𝑏𝑗𝑘  with 𝑗 = 2,3, … , 𝑛 and 𝑘 = 1,2, … , 𝑝 cannot be dominated 

by 𝑊𝑙 . 
b. If 𝑣 ∈ 𝑊𝑙  

𝑣 ∈ 𝑊𝑙 , then there exist 𝑏1𝑘 ∉ 𝑊𝑙  for 𝑘 = 1,2, … , 𝑝. Without loss of generality suppose 
that 𝑏11 ∉ 𝑊𝑙 . It means that 𝑎𝑖1 cannot be dominated by 𝑊𝑙  for 𝑖 = 2,3, … , 𝑚. 

Therefore, from two possibilities above 𝑆 is not a local dominant resolving set of  
𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝) or we can conclude that 𝑊𝑙 = 𝑝 + 1 is dominant local basis of 

𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝). Hence, we get 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) = 𝑝 + 1. ∎ 

The example of a dominant local basis for vertex amalgamation of a complete bipartite 

graph is depicted as red vertices in Figure 5, where 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐾3,3, 𝑣, 3)) = 4. 

 

 

Figure 6. 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐾3,3, 𝑣, 3)) = 4. 

 



On the Dominant Local Resolving Set of Vertex Amalgamation Graphs 

Reni Umilasari 605 

Theorem 4. Let 𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝) is a vertex amalgamation of friendship graph with the 
order is 𝑛 ≥ 3 then  

𝐷𝑑𝑖𝑚𝑙  (𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝)) = {
𝐷𝑑𝑖𝑚𝑙 (𝐹𝑛),   𝑣 𝑖𝑠 𝑎 𝑐𝑒𝑛𝑡𝑒𝑟 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 𝐹𝑛

1 + 𝑝(𝐷𝑑𝑖𝑚𝑙 (𝐹𝑛) − 1),   𝑣 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑐𝑒𝑛𝑡𝑒𝑟 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 𝐹𝑛
 

 
Proof. 
Case 1. 𝑣 is a center vertex of 𝐹𝑛  
It is very clearly to see that 𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝) ≅ 𝐹𝑛 , then by the Table 1 we can conclude that 
𝐷𝑑𝑖𝑚𝑙  (𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑝)) = 𝐷𝑑𝑖𝑚𝑙 (𝐹𝑛).  ∎ 

Case 2. 𝑣 is not a center vertex of 𝐹𝑛  
Let 𝑉(𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝)) = {𝑣, 𝑣𝑖 , 𝑥𝑖𝑘 , 𝑦𝑖𝑗 |𝑖 = 1,2,3, … , 𝑝; 𝑗 = 1,2,3, … , 𝑛; 𝑘 = 2,3,4, … , 𝑛} and 

𝐸(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑝)) = {𝑣𝑖 𝑥𝑖𝑘 , 𝑣𝑖 𝑦𝑖𝑗 , 𝑥𝑖𝑘 𝑦𝑖𝑗 , 𝑣𝑦𝑖1|𝑖 = 1,2, … , 𝑝; 𝑗 = 1,2, … , 𝑛; 𝑘 = 2,3,4, … , 𝑛 }. 

The 𝑖-th copy of 𝐹𝑛  with 𝑖 = 1,2,3, … , 𝑝 is called (𝐹𝑛)𝑖 . Let 𝐵 be a local dominant basis of 𝐹𝑛 , 
𝐵𝑖  is a local dominant basis of (𝐹𝑛 )𝑖, so that for every 𝑖 = 1,2,3, … , 𝑝, |𝐵𝑖 | = |𝐵| = 𝑛. Select 
𝑊𝑙 = {𝑣} ⋃ (𝐵𝑖 − 1)

𝑝
𝑖=1 , suppose 𝐵𝑖 − 1 = {𝑥𝑖𝑘 |𝑘 = 2,3, … , 𝑛} for every 𝑖 = 1,2,3, … , 𝑝, then 

|𝑊𝑙 | = 1 + 𝑝(𝑛 − 1).  By Lemma 2 for every 𝑥𝑎𝑏 , 𝑥𝑐𝑑 ∈ 𝐵𝑖 then 𝑟(𝑥𝑎𝑏 |𝑊𝑙 ) ≠ 𝑟(𝑥𝑐𝑑 |𝑊𝑙 ). 
Next, we take any two adjacent vertices in 𝑉(𝐺)\𝑊𝑙. Let 𝑣𝑖 , 𝑦𝑖𝑗 ∈ 𝑉(𝐺)\𝑊𝑙  with 𝑖 =

1,2,3, … , 𝑝 and 𝑗 = 2,3, … , 𝑛. Since 𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝) is a connected graph, for 𝑑(𝑦𝑖𝑗 , 𝑣) =

𝑑(𝑦𝑖𝑗 , 𝑣𝑖 ) + 𝑑(𝑣𝑖 , 𝑣) for 𝑗 ≠ 1, 𝑣 ∈ 𝑊𝑙  so that  𝑑(𝑣, 𝑦𝑖𝑗 ) ≠ 𝑑(𝑣, 𝑣𝑖 ) caused 𝑟(𝑦𝑖𝑗 |𝑊𝑙 ) ≠

𝑟(𝑣𝑖 |𝑊𝑙 ). Then, 𝑊𝑙  is local resolving set of 𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝). Moreover, since 𝑣 adjacent to 𝑣𝑖  
and 𝑥𝑖𝑘  adjacent to 𝑦𝑖𝑗 , hence 𝑊𝑙  is dominant local resolving set of 𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑝). Take any 

𝑆 ⊆ 𝑉(𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝)) with |𝑆| < |𝑊𝑙 |. Let |𝑆| = |𝑊𝑙 | − 1 the two possibilities below. 

a) If 𝑣 ∉ 𝑊𝑙  
𝑣 ∉ 𝑊𝑙 , then all vertices 𝑦𝑖1 with 𝑖 = 1,2, … , 𝑝 cannot be dominated by 𝑊𝑙 . 

b) If 𝑣 ∈ 𝑊𝑙  
𝑣 ∈ 𝑊𝑙 , then there exist 𝑥𝑖𝑘 ∉ 𝑊𝑙  for 𝑘 = 2,3 … , 𝑛. Without loss of generality suppose 
that 𝑥1𝑛 ∉ 𝑊𝑙 . It means that 𝑦1𝑛 cannot be dominated by 𝑊𝑙 . 

Therefore, from two possibilities above 𝑆 is not a local dominant resolving set of  
𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝) or we can conclude that |𝑊𝑙 | = 1 + 𝑝(𝑛 − 1) is dominant local basis of 
𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝). Hence, we get 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑝)) = 1 + 𝑝(𝐷𝑑𝑖𝑚𝑙 (𝐹𝑛) − 1) . ∎ 

 
Figure 6 gives an axample of 𝑎𝑚𝑎𝑙(𝐹3, 𝑣, 3), where 𝑣 is not the center vertex of friendship. 
Those graph has dominant local resolving set equals seven. 
 

 

Figure 7. 𝐷𝑑𝑖𝑚𝑙  (𝐹3, 𝑣, 3) = 7 
 



On the Dominant Local Resolving Set of Vertex Amalgamation Graphs 

Reni Umilasari 606 

CONCLUSION 

Based on the findings of this study, it is possible to conclude that the dominant local metric 
dimension for any vertex amalgamation product graph is determined by the dominant 
local metric dimension of the copied graphs and how the terminal vertex is chosen. This 
topic can be expanded by observing the dominant local metric dimension for the vertex 
amalgamation product with the special graphs that will be glued are different graphs. 
Next, we can determined the dominant local metric dimension for another product of 
graphs. Moreover, the program application of this concept can be generated for any 
connected graph. 

ACKNOWLEDGMENTS  

This work was supported by DRPM, KEMENRISTEK of Indonesia, Dcree 672/UN3/2022, 
Contract No. 75/UN3.15/PT/2022 and 010/E5/PG.02.00.PT/2022, year 2022. 

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