Local Multiset Dimension of Amalgamation Graphs

Ridho Alfarisi; Liliek Susilowati; Dafik Dafik; Savari Prabhu

doi:10.12688/f1000research.128866.2

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Research Article

Revised

Local Multiset Dimension of Amalgamation Graphs

[version 2; peer review: 2 approved]

Ridho Alfarisi^1,2, Liliek Susilowati ¹, Dafik Dafik³, Savari Prabhu⁴

PUBLISHED 23 Apr 2024

Author details Author details

¹ Mathematics, Universitas Airlangga, Surabaya, Surabaya, 68121, Indonesia
² Elementary School Teacher Education, Universitas Jember, Jember, East Java, 68121, Indonesia
³ Mathematics Education, Universitas Jember, Jember, East Java, 68121, Indonesia
⁴ Department of Mathematics, Rajalakshmi Engineering College, Chennai, Tamil Nadu, 602105, India

Ridho Alfarisi
Roles: Conceptualization, Investigation, Methodology, Resources, Writing – Original Draft Preparation, Writing – Review & Editing

Liliek Susilowati
Roles: Writing – Review & Editing

Dafik Dafik
Roles: Writing – Review & Editing

Savari Prabhu
Roles: Resources, Validation, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

Abstract

\textbf{Background}: One of the topics of distance in graphs is the resolving set problem. Suppose the set $W=\{s_1,s_2,…,s_k\}\subset V(G)$, the vertex representations of $\in V(G)$ is $r_m(x|W)=\{d(x,s_1),d(x,s_2),…,d(x,s_k)\}$, where $d(x,s_i)$ is the length of the shortest path of the vertex $x$ and the vertex in $W$ together with their multiplicity. The set $W$ is called a local $m$-resolving set of graphs $G$ if $r_m (v|W)\neq r_m (u|W)$ for $uv\in E(G)$. The local $m$-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of $G$, denoted by $md_l(G)$. Thus, if $G$ has an infinite local multiset dimension and then we write $md_l (G)=\infty$. \\ \textbf{Methods}: This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set $W\subset V(G)$; determine the vertex representation of two adjacent vertices in $G$; and prove the theorem.\\ \textbf{Results}: The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely $md_l(Amal(G,v,m))\leq m.md_l(G)$ and their exact value of local multiset dimension of some families of graphs namely $md_l(Amal(P_n,v,m))=1$, $md_l(Amal(K_n,v,m))=\infty$, $md_l(Amal(W_n,v,m))=m. md_l(W_n)$, $md_l(Amal(F_n,v,m))=m. md_l(F_n)$ for $d(v)=n$, $md_l(Amal(F_n,v,m))=m. \lfloor\frac{n}{4}\rfloor$. \\ \textbf{Conclusions}: We have found the upper bound of a local multiset dimension. There are some graphs which attain the upper bound of local multiset dimension namely wheel graphs\\

Keywords

local m-resolving set; local multiset dimension; amalgamation graph.

Corresponding author: Liliek Susilowati

Competing interests: No competing interests were disclosed.

Grant information: This work was supported in part by University of Jember and Universitas Airlangga, Indonesia.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Copyright: © 2024 Alfarisi R et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The author(s) is/are employees of the US Government and therefore domestic copyright protection in USA does not apply to this work. The work may be protected under the copyright laws of other jurisdictions when used in those jurisdictions.

How to cite: Alfarisi R, Susilowati L, Dafik D and Prabhu S. Local Multiset Dimension of Amalgamation Graphs [version 2; peer review: 2 approved]. F1000Research 2024, 12:95 (https://doi.org/10.12688/f1000research.128866.2) First published: 24 Jan 2023, 12:95 (https://doi.org/10.12688/f1000research.128866.1) Latest published: 23 Apr 2024, 12:95 (https://doi.org/10.12688/f1000research.128866.2)

Revised Amendments from Version 1

We have revised the introduction section for local multiset dimension notation, namely \mu(G) to md_l(G). I have revised the \Pi notation for the example to W. The notation for Figure v_0 has been revised according to the reviewer's suggestions. Writing errors have been revised according to grammar. The dot notation v has been changed to u according to the definition of the amalgamation, there is no v notation. All references have been cited in the introduction. To prove it, it has been checked for typographical errors.

See the authors' detailed response to the review by Muhammad Kamran Siddique
See the authors' detailed response to the review by Ismail Naci Cangul

Introduction

In this paper, we study the local multiset dimension of the amalgamation of graphs. One of the topics of distance in graphs is resolving set problems. This topic has many application in science and technology namely in the navigation of robots, chemistry structure, and computer sciences. The application of metric dimension in networks is the navigation of robots. Suppose, we represent a place as a vertex and a connection between place is represented by an edge. The minimum numbers of robots required to locate each vertex in the networks is part of the resolving set problems. More details of this application can be seen.¹

Several applications of resolving sets in chemistry are the substructures of a chemical compound which can be denoted by a set of functional groups. Moreover, in a chemical structure or molecular graph edges and vertices are known as bonds and atoms, respectively. Furthermore, the sub graphs are simply deliberated as substructures and functional groups. Now after altering the position of functional groups, the formed collections of compounds are distinguished as substructures being similar to each other. Later on using the method of traditional view, we can investigate if any two compounds hold the same functional group at the same point, while in drugs discovery comparative statements contributes a critical part to determine pharmacological activities related to the feature of compounds.²

All graphs G are simple, finite, and connected. Given that the vertex set V (G) and the edge set E(G), we write G = (V, E). The distance of u and v, denoted by d(u, v) is the length of the shortest path of the vertices u to v. For the set W = {s₁, s₂, …, s_k} ⊂ V (G), the vertex representations of the vertex x to the set W is an ordered k-tuple, r(x|W) = (d(x, s₁), d(x, s₂), …, d(x, s_k)). The set W is called the resolving set of G if all vertices of G have different vertex representations. The resolving set having minimum cardinality is called basis and its cardinality is called metric dimension of G, denoted by dim(G).³ Okamoto et al.⁴ introduced the new variant of resolving set problems which is called local resolving set problems. In his paper, this notion is called a local multiset dimension of graphs G. The set W is called a local resolving set if for every xy ∈ E(G), r(x|W)≠r(y|W). The local resolving set having minimum cardinality is called local basis and it’s cardinality is called local metric dimension of G and denoted by ldim(G).

Simanjuntak et al.⁵ defined multiset dimension of graphs G. Suppose the set $W = \{s_{1} s_{2} \dots s_{k}\} \subset V (G)$ , the vertex representations of a vertex x ∈ V (G) to the set W is the multiset $r_{m} (x| W) = \{d (x, s_{1}) d (x, s_{2}) \dots d (x, s_{k})\}$ where d(x, s_i) is the length of the shortest path of the vertex x and the vertex in W together with their multiplicities. The set W is called a m-resolving set if for every xy ∈ E(G), r_m(x|W)≠r_m(y|W). If G has a m-resolving set then an m-resolving set having minimum cardinality is called a multiset basis and it’s cardinality is called the multiset dimension of graphs G and denoted by md(G); otherwise we say that G has an infinite multiset dimension and we write $md (G) = \infty$ . Alfarisi et al.⁶ studied the multiset dimension of almost hypercube graphs. Later, Alfarisi et al.⁷ extended a new notion based on the multiset dimension of G, namely a local multiset dimension. Suppose the set $W = \{s_{1} s_{2} \dots s_{k}\} \subset V (G)$ , the vertex representations of a vertex x ∈ V (G) to the set W is r_m(x|W) = {d(x, s₁), d(x, s₂), …, d(x, s_k)}. The set W is called a local m-resolving set of G if r_m(v|W)≠r_m(u|W) for uv ∈ E(G). The local m-resolving set having minimum cardinality is called the local multiset basis and it’s cardinality is called the local multiset dimension, denoted by md_l(G): otherwise we say that G has an infinite local multiset dimension and we write ${md}_{l} (G) = \infty$ .

We illustrate this concept in Figure 1. In this case, the m-resolving set is W = {v₂, v₃, v₆}, 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 have distinct representations. Thus, we could have the local m-resolving set W = {v₁}, shown in Figure 1(b). Thus, the local multiset dimension is md_l(G) = 1.

r (v_{1}| W) = \{0\}, r (v_{2}| W) = \{1\}, r (v_{3}| W) = \{2\}

r (v_{4}| W) = \{1\}, r (v_{5}| W) = \{2\}, r (v_{6}| W) = \{1\}

Figure 1. (a) A graph with multiset dimension 3; (b) A graph with local multiset dimension 1.

We have some results on the local multiset dimension of some known graphs namely path, star, tree, and cycle and also the local multiset dimension of graph operations namely, cartesian product,⁷ m-shadow graph.⁸ Adawiyah et al.⁹ also studied local multiset dimension of unicyclic graphs. The followings are some results which is used for proving the new results in this study.

Lemma 0.1

Let G be a connected graphs and W ⊂ V (G). If W contains a resolving set of G, then W is a resolving set of G.¹⁰

Proposition 0.1

A graph is bipartite if and only if it contains no odd cycle.¹¹

Theorem 0.1

The local multiset dimension of G is one if and only if G is a bipartite graph.¹²

Theorem 0.2

If T is tree graph with order n, then md_l(T) = 1.¹²

Proposition 0.2

Let K_n be a complete graph with n ≥ 3, we have ${md}_{l} (K_{n}) = \infty$ .⁶

Definition 0.1

Let (G_i) be a finite collection of graphs and each G_i has a fixed vertex v called a terminal. The amalgamation Amal(G_i, v, m) is formed by taking of all the G_i and identifying their terminal.

Figure 2 is an example of an amalgamation graph with isomorphic and non-isomorphic graph.

Figure 2. a) Amal(H_j, v) with j = 1, 2, 3 and b) Amal(H₁, v, 4).

Methods

The nature of the methods follows an extrapolative design. There are several stages in this research as follows:

1. Choose the special graph which is operated by amalgamation;
2. Determining the set of vertices and edges of amalgamation of graphs;
3. Determining the set W ⊂ V (G);
4. Determining the vertex representation of two adjacent vertices in G;
5. Proving the theorem.

The flowchart of this method can be seen in Figure 3.

Figure 3. Flowchart of the extrapolate design of the study.

Results

In this section, we investigated the local multiset dimension of graph amalgamation. We provide an upper bound of local multiset dimension of Amal(G, v, m) and we show that the upper bound is sharp. We also determined the exact value of the local multiset dimension of Amal(G, v, m) for some certain graphs namely path, complete graph, fan graph, and wheel graph. The following theorem provides a sharp upper bound of Amal(G, v, m).

Lemma 0.2

Let m and n be two integers with m ≥ 2 and n ≥ 3. Let G be a connected graph of order n and v be a terminal vertex of G, then md_l (Amal(G, v, m)) ≤ m.md_l(G)

Proof.

Let

V (Amal (G, v, m)) = \{u\} \cup \{u_{j, i}; 1 \leq j \leq m; 1 \leq i \leq n - 1\}

and

E (Amal (G, v, m)) = \cup_{j = 1}^{j = m} E (G)

where v denote the identified vertex or terminal vertex.

We show that md_l (Amal(G, v, m)) ≤ m.md_l(G). Let W be the local m-resolving set of G. Amal(G, v, m) has m copies of G, such that $S = \cup_{j = 1}^{j = m} W_{j}$ . We take identified vertex v ∉ W, thus two adjacent vertices u_k,r, u_k,s ∈ V (G) have different representation. There are two conditions for u_k,r, u_k,s ∈ V (G) to terminal vertex.

1. For u_k,r, u_k,s ∈ V (G) and d (u_k,r, v) = d (u_k,s, v), then d (u_k,r, w) = d (u_k,s, w) for w ∈ S − W_k. Thus, r_m (u_k,r|S)≠r_m (u_k,s|S).
2. For u_k,r, u_k,s ∈ V (G) and d (u_k,r, v)≠d (u_k,s, v), then d (u_k,r, w)≠d (u_k,s, w) for w ∈ S − W_k. Thus, r_m (u_k,r|S)≠r_m (u_k,s|S).

Based on 1 and 2 that $S = \cup_{j = 1}^{j = m} W_{j}$ is the local m-resolving set of Amal(G, v, m). Thus, md_l (Amal(G, v, m)) ≤ m.md_l(G). □

Theorem 0.3

If Amal (P_n, v, m) is an amalgamation of m paths, then md_l (Amal(P_n, v, m)) = 1

Proof.

Let

V (Amal (P_{n}, v, m)) = \{u\} \cup \{u_{j, i}; 1 \leq j \leq m; 1 \leq i \leq n - 1\}

and

E (Amal (P_{n}, v, m)) = \cup_{j = 1}^{j = m} E ({(P_{n})}_{j})}

Since P_n is bipartite graph, then Amal (P_n, v, m) is bipartite graph. Based on Theorem 0.1 that md_l (Amal(P_n, v, m)) = 1. □

Theorem 0.4

If Amal (K_n, v, m) is an amalgamation of m complete graphs, then ${md}_{l} (Amal (K_{n}, v, m)) = \infty$

Proof.

Let

V (Amal (K_{n}, v, m)) = \{u\} \cup \{u_{j, i}; 1 \leq j \leq m; 1 \leq i \leq n - 1\}

and

E (Amal (K_{n}, v, m)) = \cup_{j = 1}^{j = m} E ({(K_{n})}_{j})}

For two adjacent vertices x, y ∈ V ((K_n)_k) with k ∈ [1, m] such that we have d(x, u) = d(y, u). Since d(x, u) = d(y, u), then d(x, w) = d(y, w) with w ∈ V (Amal (K_n, v, m)). Thus, r_m(x|W)≠r_m(y|W). □

Consider graph Amal (W_n, v, m) where v is a center vertex of W_n. Let V (Amal (W_n, v, m)) = {v} ∪ {u_{j, i};1 ≤ j ≤ m and 1 ≤ i ≤ n} such that $E (Amal (W_{n}, v, m)) = \cup_{j = 1 j = m} E ({(W_{n})}_{j})$ .

Theorem 0.5

Let m and n be two integers with m ≥ 2 and n ≥ 3. Let W_n be a wheel graph of order n + 1 and v ∈ V (W_n) where v is a center vertex of W_n, then md_l (Amal (W_n, v, m)) = m.md_l (W_n).

Proof.

Let W be a local m-resolvings of W_n and S be a local m-resolving set of Amal (W_n, v, m) Based on Lemma 0.2 that md_l (Amal(W_n, v, m)) ≤ m.md_l (W_n). Furthermore, we will show that md_l (Amal(W_n, v, m)) ≥ m.md_l (W_n). Take any S ⊂ V (Amal (W_n, v, m)) with |P|<|S|. Suppose |P|=|S| − 1 = m.|W| − 1. There is one copy of W_n, (W_n)_k have |W_k|− 1, such that we have three condition as follows

i For n ≡ 0(mod4)
Let u(k, l) ∉ P for 1 ≤ l ≤ n or i ≡ 4s − 3 with $1 \leq s \leq \frac{n}{4}$ and d (u_{k, 4(s−1)−3}, u_4(s+1)−3) > 4, then $r_{m} (u_{k, 4 (s - 1) - 1}| W_{k}) = \dots = r_{m} (u_{4 (s + 1) - 5}| W_{k}) = \underset{\frac{n - 4}{4}}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 4 (s - 1) - 1}, w) = \dots = d (u_{k, 4 (s + 1) - 5}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 4 (s - 1) - 1}| P) = \dots = r_{m} (u_{k, 4 (s + 1) - 5}| P)$ , then P is not local m-resolving set of Amal(W_n, v, m).
ii For n ≡ 2(mod4)
There are three possibilities namely 1) u_k,1 ∉ P; 2) u_k,l ∉ P for 1 < l < n − 1 and 3) u_k,n−1 ∈ P.
- (a) Let u_k,1 ∉ P, then $r_{m} (u_{k, 1}| W_{k}) = \dots = r_{m} (u_{k, 3}| W_{k}) = \underset{⌈\frac{n}{4}⌉ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 1}, w) = \dots = d (u_{k, 3}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 1}| P) = \dots = r_{m} (u_{k, 3}| P)$ , then P is not local m-resolving set of Amal (W_n, v, m).
- (b) Let u_k,l ∉ P for 1 < l < n − 1 or i ≡ 4s − 3 with $2 \leq k \leq ⌈\frac{n}{4}⌉ - 1$ and d (u_{k, 4(s−1)−3}, u_{k, 4(s+1)−3}) > 4, then $r_{m} (u_{k, 4 (s - 1) - 1}| W_{k}) = \dots = r_{m} (u_{k, 4 (s + 1) - 5}| W_{k}) = \underset{⌈\frac{n}{4}⌉ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 4 (s - 1) - 1}, w) = \dots = d (u_{k, 4 (s + 1) - 5}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 4 (s - 1) - 1}| P) = \dots = r_{m} (u_{k, 4 (s + 1) - 5}| P)$ , then P is not local m-resolving set of Amal (W_n, v, m).
- (c) Let u_k,n−1 ∉ P. Since $r_{m} (u_{k, n - 3}| W_{k}) = \dots = r_{m} (u_{k, n - 1}| W_{k}) = \underset{⌈\frac{n}{4}⌉ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, n - 3}, w) = \dots = d (u_{k, n - 1}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, n - 3}| P) = \dots = r_{m} (u_{k, n - 1}| P)$ , then P is not local m-resolving set of Amal (W_n, v, m).
iii For n ≡ 1, 3(mod4)
We know that ${md}_{l} (W_{n}) = \infty$ , so there are two adjacent vertices x, y ∈ V (W_n), r_m(x|W_k) = r_m(y|W_k). Since d(x, v) = d(y, v) and d(x, w) = d(x, w) where w ∈ P − W_k, then r_m(x|P) = r_m(y|P). Thus, ${md}_{l} (Amal (W_{n}, v, m)) = \infty$ .

Based on i), ii), and iii) that md_l (Amal (W_n, v, m)) = m.md_l (W_n). □

For further discussion, consider graph Amal (W_n, v, m) where v is not a center vertex of W_n. Let V (Amal (W_n, v, m)) = {v} ∪ {u_j;1 ≤ j ≤ m} ∪ {u_{j, i};1 ≤ j ≤ m and 1 ≤ i ≤ n − 1} such that E (Amal(W_n, v, m)) = {vu_j, vu_j,1, vu_j,n−1}∪{u_{j, i}u_{j, i+1};1 ≤ i ≤ n−2}∪{u_ju_{j, i};1 ≤ j ≤ m, 1 ≤ i ≤ n−1}.

Theorem 0.6

Let m and n be two integers with m ≥ 2 and n ≥ 3. Let W_n be a wheel graph of order n + 1 and v ∈ V (W_n) where v is not center vertex of W_n, then

{md}_{l} (Amal (W_{n}, v, m)) = \{\begin{cases} m . (\frac{n}{4} - 1), & for n \equiv 0 (mod 4) \\ m . ⌊ \frac{n}{4} ⌋, & for n \equiv 1, 2, 3 (mod 4) \end{cases}

Proof.

We consider four cases.

Case 1.

For n ≡ 0(mod 4)

Choose S = {u_j,i;1 ≤ j ≤ m, 1 ≤ i ≤ n − 1, i ≡ 0(mod4)} obtained the vertex representation as follows

1. For u_k, v ∈ V (Amal (W_n, v, m)), d(v, w)≠d (u_k, w) where w ∈ S − W_k such that r_m(v|S − W_k)≠r_m (u_k|S − W_k). Since d(v, w)≠d (u_k, w) where w ∈ W_k such that r_m(v|W_k)≠r_m (u_k|W_k). Thus, r_m(v|S)≠r_m (u_k|S).
2. For u_k, u_k,l ∈ V ((W_n)_k) and l≠0(mod4). Since u_k adjacent to w where w ∈ W_k, such that r_m (u_k|W_k)≠r_m (u_k,l|W_k). Since d (u_k, w)≠d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k)≠r_m (u_k,l|S − W_k) with l≠1, n − 1. Since d (u_k, w) = d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k) = r_m (u_k,l|S − W_k) with l = 1, n − 1. Thus, r_m(v|S)≠r_m (u_k|S).
3. For u_{k, i} ∈ V ((W_n)_k) and 1 ≤ i ≤ 3. Since d (u_k,1, w)≠d (u_k,2, w) where w ∈ S − W_k and d (u_k,1, w)≠d (u_k,2, w) where w ∈ W_k such that r_m (u_k,1|S)≠r_m (u_k,2|S). Since d (u_k,2, w) = d (u_k,3, w) where w ∈ S − W_k and d (u_k,2, w)≠d (u_k,3, w) where w ∈ W_k such that r_m (u_k,2|S)≠r_m (u_k,3|S).
4. For u_{k, i} ∈ V ((W_n)_k) and n − 3 ≤ i ≤ n − 1. Since d (u_k,n−1, w)≠d (u_k,n−2, w) where w ∈ S − W_k and d (u_k,n−1, w)≠d (u_k,n−2, w) where w ∈ W_k such that r_m (u_k,n−1|S)≠r_m (u_k,n−2|S). Since d (u_k,n−2, w) = d (u_k,n−3, w) where w ∈ S − W_k and d (u_k,n−2, w)≠d (u_k,n−3, w) where w ∈ W_k such that r_m (u_k,n−2|S)≠r_m (u_k,n−3|S).
5. For u_{k, i} ∈ V ((W_n)_k) where 4l + 1 ≤ i ≤ 4l + 3 and $1 \leq l \leq \frac{n - 8}{4}$ . Since d (u_k,4l+1, w)≠d (u_k,4l+2, w) where w ∈ S − W_k and d (u_k,4l+1, w)≠d (u_k,4l+2, w) where w ∈ W_k such that r_m (u_k,4l+1|S)≠r_m (u_k,4l+2|S). Since d (u_k,4l+2, w) = d (u_k,4l+3, w) where w ∈ S − W_k and d (u_k,4l+2, w)≠d (u_k,4l+3, w) where w ∈ W_k such that r_m (u_k,4l+2|S)≠r_m (u_{k, 4l+3}|S).

Based on the representation above that every two adjacent vertices has distinct representations such that S is local m-resolving set and ${md}_{l} (Amal (W_{n}, v, m)) \leq m . (\frac{n}{4} - 1)$ . Furthermore, proving that m ${md}_{l} (Amal (W_{n}, v, m)) \geq m . (\frac{n}{4} - 1)$ . Taking any set P ⊂ V (Amal (W_n, v, m)) with $| P | = m . (\frac{n}{4} - 1) - 1$ . Let u(k, l) ∉ P for l ≡ 4s with $1 \leq s \leq \frac{n - 8}{4}$ and d (u_{k, 4(s−1)}, u_k,4(s+1)) > 4, then $r_{m} (u_{k, 4 (s - 1) + 2}| W_{k}) = \dots = r_{m} (u_{4 (s + 1) - 2}| W_{k}) = \underset{\frac{n - 8}{4}}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 4 (s - 1) + 2}, w) = \dots = d (u_{k, 4 (s + 1) - 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 4 (s - 1) + 2}| P) = \dots = r_{m} (u_{k, 4 (s + 1) - 2}| P)$ , then P is not local m-resolving set of Amal (W_n, v, m). Thus, ${md}_{l} (Amal (W_{n}, v, m)) = m . (\frac{n}{4} - 1)$ .

Case 2.

For n ≡ 1(mod 4)

Choose S = {u_{j, i};1 ≤ j ≤ m, 1 ≤ i ≤ n − 1, i ≡ 0(mod4)} obtained the vertex representation as follows

1. For u_k, v ∈ V (Amal (W_n, v, m)), d(v, w)≠d (u_k, w) where w ∈ S − W_k such that r_m(v|S − W_k)≠r_m (u_k|S − W_k). Since d(v, w)≠d (u_k, w) where w ∈ W_k such that r_m(v|W_k)≠r_m (u_k|W_k). Thus, r_m(v|S)≠r_m (u_k|S).
2. For u_k, u_k,l ∈ V ((W_n)_k) and l≠0(mod4). Since u_k adjacent to w where w ∈ W_k, such that r_m (u_k|W_k)≠r_m (u_k,l|W_k). Since d (u_k, w)≠d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k)≠r_m (u_k,l|S − W_k) with l≠1, n − 1. Since d (u_k, w) = d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k) = r_m (u_k,l|S − W_k) with l = 1, n − 1. Thus, r_m(v|S)≠r_m (u_k|S).
3. For u_{k, i} ∈ V ((W_n)_k) and 1 ≤ i ≤ 3. Since d (u_k,1, w)≠d (u_k,2, w) where w ∈ S − W_k and d (u_k,1, w)≠d (u_k,2, w) where w ∈ W_k such that r_m (u_k,1|S)≠r_m (u_k,2|S). Since d (u_k,2, w) = d (u_k,3, w) where w ∈ S − W_k and d (u_k,2, w)≠d (u_k,3, w) where w ∈ W_k such that r_m (u_k,2|S)≠r_m (u_k,3|S).
4. For u_{k, i} ∈ V ((W_n)_k) where 4l + 1 ≤ i ≤ 4l + 3 and $1 \leq l \leq \frac{n - 5}{4}$ . Since d (u_k,4l+1, w)≠d (u_k,4l+2, w) where w ∈ S − W_k and d (u_k,4l+1, w)≠d (u_k,4l+2, w) where w ∈ W_k such that r_m (u_k,4l+1|S)≠r_m (u_k,4l+2|S). Since d (u_k,4l+2, w) = d (u_k,4l+3, w) where w ∈ S − W_k and d (u_k,4l+2, w)≠d (u_k,4l+3, w) where w ∈ W_k such that r_m (u_k,4l+2|S)≠r_m (u_k,4l+3|S).

Based on the representation above that every two adjacent vertices has distinct representation such that S is local m-resolving set and ${md}_{l} (Amal (W_{n}, v, m)) \leq m . (\frac{n - 1}{4})$ . Furthermore, proving that m ${md}_{l} (Amal (W_{n}, v, m)) \geq m . (\frac{n - 1}{4})$ . Taking any set P ⊂ V (Amal (W_n, v, m)) with $| P | = m . (\frac{n - 1}{4}) - 1$ . Let u₍k, l) ∉ P for l ≡ 4s with $1 \leq s \leq \frac{n - 5}{4}$ and d (u_k,4(s−1), u_k,4(s+1)) > 4, then $r_{m} (u_{k, 4 (s - 1) + 2}| W_{k}) = \dots = r_{m} (u_{4 (s + 1) - 2}| W_{k}) = \underset{\frac{n - 5}{4}}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 4 (s - 1) + 2}, w) = \dots = d (u_{k, 4 (s + 1) - 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 4 (s - 1) + 2}| P) = \dots = r_{m} (u_{k, 4 (s + 1) - 2}| P)$ , then P is not local m-resolving set of Amal (W_n, v, m). Thus, ${md}_{l} (Amal (W_{n}, v, m)) = m . (\frac{n - 1}{4})$

Case 3.

For n ≡ 2(mod 4)

Choose S = {u_{j, i};1 ≤ j ≤ m, 1 ≤ i ≤ n − 1, i ≡ 0(mod4)} obtained the vertex representation as follows

1. For u_k, v ∈ V (Amal (W_n, v, m)), d(v, w)≠d (u_k, w) where w ∈ S − W_k such that r_m(v|S − W_k)≠r_m (u_k|S − W_k). Since d(v, w)≠d (u_k, w) where w ∈ W_k such that r_m(v|W_k)≠r_m (u_k|W_k). Thus, r_m(v|S)≠r_m (u_k|S).
2. For u_k, u_k,l ∈ V ((W_n)_k) and l≠0(mod4). Since u_k adjacent to w where w ∈ W_k, such that r_m (u_k|W_k)≠r_m (u_k,l|W_k). Since d (u_k, w)≠d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k)≠r_m (u_k,l|S − W_k) with l≠1, n − 1. Since d (u_k, w) = d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k) = r_m (u_k,l|S − W_k) with l = 1, n − 1. Thus, r_m(v|S)≠r_m (u_k|S).
3. For u_k,i ∈ V ((W_n)_k) and 1 ≤ i ≤ 3. Since d (u_k,1, w)≠d (u_k,2, w) where w ∈ S − W_k and d (u_k,1, w)≠d (u_k,2, w) where w ∈ W_k such that r_m (u_k,1|S)≠r_m (u_k,2|S). Since d (u_k,2, w) = d (u_k,3, w) where w ∈ S − W_k and d (u_k,2, w)≠d (u_k,3, w) where w ∈ W_k such that r_m (u_k,2|S)≠r_m (u_k,3|S).
4. For u_{k, i} ∈ V ((W_n)_k) where 4l + 1 ≤ i ≤ 4l + 3 and $1 \leq l \leq \frac{n - 6}{4}$ . Since d (u_k,4l+1, w)≠d (u_k,4l+2, w) where w ∈ S − W_k and d (u_k,4l+1, w)≠d (u_k,4l+2, w) where w ∈ W_k such that r_m (u_k,4l+1|S)≠r_m (u_k,4l+2|S). Since d (u_k,4l+2, w) = d (u_k,4l+3, w) where w ∈ S − W_k and d (u_k,4l+2, w)≠d (u_k,4l+3, w) where w ∈ W_k such that r_m (u_k,4l+2|S)≠r_m (u_k,4l+3|S).

Based on the representation above that every two adjacent vertices has distinct representation such that S is local m-resolving set and ${md}_{l} (Amal (W_{n}, v, m)) \leq m . (\frac{n - 2}{4})$ . Furthermore, proving that m ${md}_{l} (Amal (W_{n}, v, m)) \geq m . (\frac{n - 2}{4})$ . Taking any set P ⊂ V (Amal (W_n, v, m)) with $| P | = m . (\frac{n - 2}{4}) - 1$ . Let u₍k, l) ∉ P for l ≡ 4s with $1 \leq s \leq \frac{n - 6}{4}$ and d (u_k,4(s−1), u_k,4(s+1)) > 4, then $r_{m} (u_{k, 4 (s - 1) + 2}| W_{k}) = \dots = r_{m} (u_{4 (s + 1) - 2}| W_{k}) = \underset{\frac{n - 6}{4}}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 4 (s - 1) + 2}, w) = \dots = d (u_{k, 4 (s + 1) - 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 4 (s - 1) + 2}| P) = \dots = r_{m} (u_{k, 4 (s + 1) - 2}| P)$ , then P is not local m-resolving set of Amal (W_n, v, m). Thus, ${md}_{l} (Amal (W_{n}, v, m)) = m . (\frac{n - 2}{4})$

Case 4.

For n ≡ 3(mod 4)

Choose S = {u_{j, i};1 ≤ j ≤ m, 1 ≤ i ≤ n − 1, i ≡ 0(mod4)} obtained the vertex representation as follows

1. For u_k, v ∈ V (Amal (W_n, v, m)), d(v, w)≠d (u_k, w) where w ∈ S − W_k such that r_m(v|S − W_k)≠r_m (u_k|S − W_k). Since d(v, w)≠d (u_k, w) where w ∈ W_k such that r_m(v|W_k)≠r_m (u_k|W_k). Thus, r_m(v|S)≠r_m (u_k|S).
2. For u_k, u_k,l ∈ V ((W_n)_k) and l≠0(mod4). Since u_k adjacent to w where w ∈ W_k, such that r_m (u_k|W_k)≠r_m (u_k,l|W_k). Since d (u_k, w)≠d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k)≠r_m (u_k,l|S − W_k) with l≠1, n − 1. Since d (u_k, w) = d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k) = r_m (u_k,l|S − W_k) with l = 1, n − 1. Thus, r_m(v|S)≠r_m (u_k|S).
3. For u_{k, i} ∈ V ((W_n)_k) and 1 ≤ i ≤ 3. Since d (u_k,1, w)≠d (u_k,2, w) where w ∈ S − W_k and d (u_k,1, w)≠d (u_k,2, w) where w ∈ W_k such that r_m (u_k,1|S)≠r_m (u_k,2|S). Since d (u_k,2, w) = d (u_k,3, w) where w ∈ S − W_k and d (u_k,2, w)≠d (u_k,3, w) where w ∈ W_k such that r_m (u_k,2|S)≠r_m (u_k,3|S).
4. For u_{k, i} ∈ V ((W_n)_k) and n − 2 ≤ i ≤ n − 1. Since d (u_k,n−1, w)≠d (u_k,n−2, w) where w ∈ S − W_k and d (u_k,n−1, w)≠d (u_k,n−2, w) where w ∈ W_k such that r_m (u_k,n−1|S)≠r_m (u_k,n−2|S).
5. For u_{k, i} ∈ V ((W_n)_k) where 4l + 1 ≤ i ≤ 4l + 3 and $1 \leq l \leq \frac{n - 7}{4}$ . Since d (u_k,4l+1, w)≠d (u_k,4l+2, w) where w ∈ S − W_k and d (u_k,4l+1, w)≠d (u_k,4l+2, w) where w ∈ W_k such that r_m (u_k,4l+1|S)≠r_m (u_k,4l+2|S). Since d (u_k,4l+2, w) = d (u_k,4l+3, w) where w ∈ S − W_k and d (u_k,4l+2, w)≠d (u_k,4l+3, w) where w ∈ W_k such that r_m (u_k,4l+2|S)≠r_m (u_k,4l+3|S).

Based on the representation above that every two adjacent vertices has distinct representations such that S is local m-resolving set and ${md}_{l} (Amal (W_{n}, v, m)) \leq m . (\frac{n - 3}{4})$ . Furthermore, proving that m ${md}_{l} (Amal (W_{n}, v, m)) \geq m . (\frac{n - 3}{4})$ . Taking any set P ⊂ V (Amal (W_n, v, m)) with $| P | = m . (\frac{n - 3}{4}) - 1$ . Let u₍k, l) ∉ P for l ≡ 4s with $1 \leq s \leq \frac{n - 7}{4}$ and d (u_k,4(s−1), u_{k, 4(s+1)}) > 4, then $r_{m} (u_{k, 4 (s - 1) + 2}| W_{k}) = \dots = r_{m} (u_{4 (s + 1) - 2}| W_{k}) = \underset{\frac{n - 7}{4}}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 4 (s - 1) + 2}, w) = \dots = d (u_{k, 4 (s + 1) - 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 4 (s - 1) + 2}| P) = \dots = r_{m} (u_{k, 4 (s + 1) - 2}| P)$ , then P is not local m-resolving set of Amal (W_n, v, m). Thus, ${md}_{l} (Amal (W_{n}, v, m)) = m . (\frac{n - 3}{4})$ . □

Consider graph Amal (F_n, v, m) where v is a center vertex of F_n. Let V (Amal (F_n, v, m)) = {v} ∪ {u_{j, i};1 ≤ j ≤ mand1 ≤ i ≤ n} such that $E (Amal (F_{n}, v, m)) = \cup_{j = 1 j = m} E ({(F_{n})}_{j})$ .

Theorem 0.7

Let m and n be two integers with m ≥ 2 and n ≥ 6. Let F_n be a fan graph of order n + 1 and v ∈ V (F_n) where v is a center vertex of F_n, then md_l (Amal (F_n, v, m)) = m.md_l (F_n)

Proof.

Let W be a local m-resolving set of F_n and S be a local m-resolving set of Amal (F_n, v, m) Based on Lemma 0.2 that md_l (Amal(F_n, v, m)) ≤ m.md_l (F_n). Furthermore, we will show that md_l (Amal(F_n, v, m)) ≥ m.md_l (F_n). Take any S ⊂ V (Amal (F_n, v, m)) with |P|<|S|. Suppose |P|=|S| − 1 = m.|W| − 1. There are one copies of F_n, (F_n)_k have |W_k|− 1, such that we have two condition as follows

i For n ≡ 1(mod4)
There are three possibilities namely 1) u_k,3 ∉ P; 2) u_k,l ∉ P for 3 < l < n − 2 and 3) u_k,n−2 ∈ P.
- (a) Let u_k,3 ∉ P. Since u_k,1u_k,2 ∈ E((F_n)_k), then $r_{m} (u_{k, 1}| W_{k}) = r_{m} (u_{k, 2}| W_{k}) = \underset{\frac{n - 5}{4}}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since d (u_k,1, w) = d (u_k,2, w) for w ∈ P − W_k and r_m (u_k,1|P) = r_m (u_k,2|P), then P is not local m-resolving set of Amal (F_n, v, m).
- (b) Let u_k,l ∉ P for 3 < l < n − 2 or i ≡ 4s − 1 with $2 \leq s \leq \frac{n - 5}{4}$ and d (u_{k, 4(s−1)−1}, u_{k, 4(s+1)−1}) > 4, then $r_{m} (u_{k, 4 (s - 1) + 1}| W_{k}) = \dots = r_{m} (u_{k, 4 (s + 1) - 3}| W_{k}) = \underset{\frac{n - 5}{4}}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 4 (s - 1) + 1}, w) = \dots = d (u_{k, 4 (s + 1) - 3}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 4 (s - 1) + 1}| P) = \dots = r_{m} (u_{k, 4 (s + 1) - 3}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
- (c) Let u_k,n−2 ∉ P. Since u_k,n−1u_{k, n} ∈ E((F_n)_k), then $r_{m} (u_{k, n - 4}| W_{k}) = \dots = r_{m} (u_{k, n}| W_{k}) = \underset{\frac{n - 5}{4}}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, n - 4}, w) = \dots = d (u_{k, n}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, n - 4}| P) = \dots = r_{m} (u_{k, n}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
ii For n ≡ 0, 2, 3(mod4)
There are three possibilities namely 1) u_k,1 ∉ P; 2) u_k,l ∉ P for 1 < l < n − 1 and 3) u_k,n−1 ∈ P for n ≡ 0(mod4), u_k,n−1 ∈ P for n ≡ 2(mod4), and u_k,n ∈ P for n ≡ 3(mod4).
- (a) Let u_k,3 ∉ P. Since $r_{m} (u_{k, 1}| W_{k}) = \dots = r_{m} (u_{k, 5}| W_{k}) = \underset{⌈\frac{n}{4}⌉ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 1}, w) = \dots = d (u_{k, 5}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 1}| P) = \dots = r_{m} (u_{k, 5}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
- (b) Let u_k,l ∉ P for 3 < l < n − 2 or i ≡ 4s − 1 with $2 \leq s \leq ⌈\frac{n}{4}⌉$ and d (u_{k,4(s−1)−1}, u_{k, 4(s+1)−1}) > 4, then $r_{m} (u_{k, 4 (s - 1) + 1}| W_{k}) = \dots = r_{m} (u_{k, 4 (s + 1) - 3}| W_{k}) = \underset{⌈\frac{n}{4}⌉ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 4 (s - 1) + 1}, w) = \dots = d (u_{k, 4 (s + 1) - 3}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 4 (s - 1) + 1}| P) = \dots = r_{m} (u_{k, 4 (s + 1) - 3}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
- (c) Let u_k,n−1 ∉ P for n ≡ 0(mod4). Since $r_{m} (u_{k, n - 3}| W_{k}) = \dots = r_{m} (u_{k, n}| W_{k}) = \underset{⌈\frac{n}{4}⌉ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, n - 3}, w) = \dots = d (u_{k, n}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, n - 3}| P) = \dots = r_{m} (u_{k, n}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
- (d) Let u_k,n−1 ∉ P for n ≡ 2(mod4). Since $r_{m} (u_{k, n - 1}| W_{k}) = r_{m} (u_{k, n}| W_{k}) = \underset{⌈\frac{n}{4}⌉ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since d (u_k,n−1, w) = d (u_k,n, w) for w ∈ P − W_k and r_m (u_k,n−1|P) = r_m (u_k,n|P), then P is not local m-resolving set of Amal (F_n, v, m).
- (e) Let u_k,n ∉ P for n ≡ 3(mod4). Since $r_{m} (u_{k, n - 2}| W_{k}) = \dots = r_{m} (u_{k, n}| W_{k}) = \underset{⌈\frac{n}{4}⌉ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, n - 2}, w) = \dots = d (u_{k, n}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, n - 2}| P) = \dots = r_{m} (u_{k, n}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).

Based on i) and ii) that md_l (Amal (F_n, v, m)) = m.md_l (F_n). □

For further discussion, consider graph Amal (F_n, v, m) where v ∈ V (F_n) with d(v) = 2. Let V (Amal (F_n, v, m)) = {v} ∪ {u_j;1 ≤ j ≤ m} ∪ {u_{j, i};1 ≤ j ≤ mand1 ≤ i ≤ n − 1} such that E (Amal(F_n, v, m)) = {vu_j, vu_j,1}∪{u_{j, i}u_{j, i+1};1 ≤ i ≤ n−2}∪{u_ju_{j, i};1 ≤ j ≤ m, 1 ≤ i ≤ n−1}.

Theorem 0.8

Let m and n be two integers with m ≥ 2 and n ≥ 3. Let F_n be a fan graph of order n + 1 and v ∈ V (F_n) where d(v) = 2, then ${md}_{l} (Amal (F_{n}, v, m)) = m . ⌊\frac{n}{4}⌋$

Proof.

We have some condition for the set S as follows

a for n ≡ 1, 2, 3(mod4), S = {u_{j, i};1 ≤ j ≤ m, 1 ≤ i ≤ n − 1, i ≡ 0(mod4)}.
b for n ≡ 0(mod4), S = {u_{j, i};1 ≤ j ≤ m, 1 ≤ i ≤ n − 1, i ≡ 0(mod4)} ∪ {u_j,n−2}.

We obtained the vertex representation as follows

1. For u_k, v ∈ V (Amal (F_n, v, m)), d(v, w)≠d (u_k, w) where w ∈ S − W_k such that r_m(v|S − W_k)≠r_m (u_k|S − W_k). Since d(v, w)≠d (u_k, w) where w ∈ W_k such that r_m(v|W_k)≠r_m (u_k|W_k). Thus, r_m(v|S)≠r_m (u_k|S).
2. For u_k, u_k,l ∈ V ((F_n)_k) where l≠0(mod4) and u_k,l where l≠n − 2 for n ≡ 0(mod4). Since u_k is adjacent to w where w ∈ W_k, such that r_m (u_k|W_k)≠r_m (u_k,l|W_k). Since d (u_k, w)≠d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k)≠r_m (u_k,l|S − W_k) with l≠1. Since d (u_k, w) = d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k) = r_m (u_k,l|S − W_k) with l = 1. Thus, r_m(v|S)≠r_m (u_k|S).
3. For u_{k, i} ∈ V ((F_n)_k) and 1 ≤ i ≤ 3. Since d (u_k,1, w)≠d (u_k,2, w) where w ∈ S − W_k and d (u_k,1, w)≠d (u_k,2, w) where w ∈ W_k such that r_m (u_k,1|S)≠r_m (u_k,2|S). Since d (u_k,2, w) = d (u_k,3, w) where w ∈ S − W_k and d (u_k,2, w)≠d (u_k,3, w) where w ∈ W_k such that r_m (u_k,2|S)≠r_m (u_k,3|S).
4. For u_{k, i} ∈ V ((F_n)_k) where 4l + 1 ≤ i ≤ 4l + 3 and $1 \leq l \leq ⌊\frac{n}{4}⌋ - 1$ . Since d (u_k,4l+1, w)≠d (u_k,4l+2, w) where w ∈ S − W_k and d (u_k,4l+1, w)≠d (u_k,4l+2, w) where w ∈ W_k such that r_m (u_k,4l+1|S)≠r_m (u_k,4l+2|S). Since d (u_k,4l+2, w) = d (u_k,4l+3, w) where w ∈ S − W_k and d (u_k,4l+2, w)≠d (u_k,4l+3, w) where w ∈ W_k such that r_m (u_k,4l+2|S)≠r_m (u_k,4l+3|S).
5. For u_{k, i} ∈ V ((F_n)_k) where n − 2 ≤ i ≤ n − 1 and n ≡ 3(mod4). Since d (u_k,n−1, w) = d (u_k,n−2, w) where w ∈ S − W_k and d (u_k,n−1, w)≠d (u_k,n−2, w) where w ∈ W_k such that r_m (u_k,n−1|S)≠r_m (u_k,n−2|S).

Based on the representation above that every two adjacent vertices has distinct representations such that S is local m-resolving set and ${md}_{l} (Amal (W_{n}, v, m)) \leq m . (\frac{n}{4} - 1)$ .

Furthermore, we will show that ${md}_{l} (Amal (F_{n}, v, m)) \geq m . ⌊\frac{n}{4}⌋$ . Take any S ⊂ V (Amal (F_n, v, m)) with|P|<|S|. Suppose $| P | = | S | - 1 = m . ⌊\frac{n}{4}⌋ - 1$ . There are one copies of F_n, (F_n)_k have $⌊\frac{n}{4}⌋ - 1$ , such that we have two condition as follows

i For n ≡ 1(mod4)
There are two possibilities namely 1) u_k,l ∉ P for 1 < l < n − 1 and 2) u_k,n−1 ∉ P.
- (a) Let u_k,l ∉ P for 1 < l ≤ n − 1 or l ≡ 4s with $1 \leq s \leq \frac{n - 5}{4}$ and d (u_{k, 4(s−1)+1}, u_{k, 4(s+1)−1}) > 4, then $r_{m} (u_{k, 4 (s - 1) + 2}| W_{k}) = \dots = r_{m} (u_{k, 4 (s + 1) - 2}| W_{k}) = \underset{\frac{n - 5}{4}}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 4 (s - 1) + 2}, w) = \dots = d (u_{k, 4 (s + 1) - 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 4 (s - 1) + 2}| P) = \dots = r_{m} (u_{k, 4 (s + 1) - 2}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
- (b) Let u_k,n−1 ∉ P. Since $r_{m} (u_{k, n - 3}| W_{k}) = \dots = r_{m} (u_{k, n - 1}| W_{k}) = \underset{\frac{n - 5}{4}}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, n - 3}, w) = \dots = d (u_{k, n - 1}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, n - 3}| P) = \dots = r_{m} (u_{k, n - 1}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
ii For n ≡ 0, 2, 3(mod4)
There are three possibilities namely 1) u_k,l ∉ P for 1 < l < n − 1 and 2) u_k,n−1 ∈ P for n ≡ 0(mod4), u_k,n−2 ∈ P for n ≡ 2(mod4), and u_k,n−3 ∈ P for n ≡ 3(mod4).
- (a) Let u_k,l ∉ P for 1 < l < n − 1 or l ≡ 4s with $1 \leq s \leq ⌊\frac{n}{4}⌋ - 1$ and d (u_{k, 4(s−1)+1}, u_{k, 4(s+1)−1}) > 4, then $r_{m} (u_{k, 4 (s - 1) + 2}| W_{k}) = \dots = r_{m} (u_{k, 4 (s + 1) - 2}| W_{k}) = \underset{⌊\frac{n}{4}⌋ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, 4 (s - 1) + 2}, w) = \dots = d (u_{k, 4 (s + 1) - 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, 4 (s - 1) + 2}| P) = \dots = r_{m} (u_{k, 4 (s + 1) - 2}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
- (b) Let u_k,n−1 ∉ P for n ≡ 0(mod4). Since $r_{m} (u_{k, n - 2}| W_{k}) = r_{m} (u_{k, n - 1}| W_{k}) = \underset{⌊\frac{n}{4}⌋ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since d (u_k,n−2, w) = d (u_k,n−1, w) for w ∈ P − W_k and r_m (u_k,n−2|P) = r_m (u_k,n−1|P), then P is not local m-resolving set of Amal (F_n, v, m).
- (c) Let u_k,n−2 ∉ P for n ≡ 2(mod4). Since $r_{m} (u_{k, n - 4}| W_{k}) = \dots = r_{m} (u_{k, n - 1}| W_{k}) = \underset{⌊\frac{n}{4}⌋ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, n - 4}, w) = \dots = d (u_{k, n - 1}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, n - 4}| P) = \dots = r_{m} (u_{k, n - 1}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
- (d) Let u_k,n−3 ∉ P for n ≡ 3(mod4). Since $r_{m} (u_{k, n - 5}| W_{k}) = \dots = r_{m} (u_{k, n - 1}| W_{k}) = \underset{⌈\frac{n}{4}⌉ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, n - 5}, w) = \dots = d (u_{k, n - 1}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, n - 5}| P) = \dots = r_{m} (u_{k, n - 1}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).

Based on i) and ii) that ${md}_{l} (Amal (F_{n}, v, m)) = m . ⌊\frac{n}{4}⌋$ . □

Theorem 0.9

Let m and n be two integers with m ≥ 2 and n ≥ 3. Let F_n be a fan graph of order n + 1 and v ∈ V (F_n) where d(v) = 3, then

{md}_{l} (Amal (F_{n}, v, m)) = \{\begin{cases} m . (⌊\frac{p}{4}⌋ + ⌊\frac{n - p}{4}⌋), & for n - p \equiv 0, 1, 2 (mod 4) and 2 \leq p \leq ⌈\frac{n}{4}⌉ \\ m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉), & for n - p \equiv 3 (mod 4) and 2 \leq p \leq ⌈\frac{n}{4}⌉ \end{cases}

Proof.

Since Identified vertex v = u_{j, p}, then we have two part namely u_j,1 − u_{j, p−1} and u_{j, p+1} − u_{j, n}. We consider two cases as follows

Case 1.

For 2 ≤ p ≤ 3

For d (u_j,1, u_{j, p−1}) ≤ 2, then the vertices $u_{j, 1}, \dots, u_{j, p - 1}$ don’t need to be resolver. Thus, we focus for d (u_{j, p+1}, u_{j, n}) ≥ 3

Sub Case 1.1.

For n − p ≡ 0, 1, 2(mod4)

Choose S = {u_j,r;1 ≤ j ≤ m, r ≡ p (mod4), r > p} obtained the vertex representation as follows

1. For u_k,l ∈ V (Amal (F_n, v, m)) and 1 ≤ l ≤ p − 1, d (u_k,1, w)≠d (u_k,2, w) where w ∈ S − W_k such that r_m (u_k,1|S − W_k)≠r_m (u_k,2|S − W_k). Since d (u_k,1, w) = d (u_k,2, w) where w ∈ W_k such that r_m (u_k,1|W_k)≠r_m (u_k,2|W_k). Thus, r_m (u_k,1|S)≠r_m (u_k,2|S).
2. For u_k, v ∈ V (Amal (F_n, v, m)), d(v, w)≠d (u_k, w) where w ∈ S − W_k such that r_m(v|S − W_k)≠r_m (u_k|S − W_k). Since d(v, w)≠d (u_k, w) where w ∈ W_k such that r_m(v|W_k)≠r_m (u_k|W_k). Thus, r_m(v|S)≠r_m (u_k|S).
3. For u_k, u_k,l ∈ V ((F_n)_k) where l≠p (mod4) and u_k,l where l≠n − 2 for n − p ≡ 2(mod4). Since u_k adjacent to w where w ∈ W_k, such that r_m (u_k|W_k)≠r_m (u_k,l|W_k). Since d (u_k, w)≠d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k)≠r_m (u_k,l|S − W_k) with l≠1. Since d (u_k, w) = d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k) = r_m (u_k,l|S − W_k) with l = 1. Thus, r_m(v|S)≠r_m (u_k|S).
4. For u_{k, i} ∈ V ((F_n)_k) and p + 1 ≤ i ≤ p + 3. Since d (u_{k, p+1}, w)≠d (u_{k, p+2}, w) where w ∈ S − W_k and d (u_{k, p+1}, w)≠d (u_{k, p+2}, w) where w ∈ W_k such that r_m (u_{k, p+1}|S)≠r_m (u_{k, p+2}|S). Since d (u_{k, p+2}, w) = d (u_{k, p+3}, w) where w ∈ S − W_k and d (u_{k, p+2}, w)≠d (u_{k, p+3}, w) where w ∈ W_k such that r_m (u_{k, p+2}|S)≠r_m (u_{k, p+3}|S).
5. For u_{k, i} ∈ V ((F_n)_k) where p + 4l + 1 ≤ i ≤ p + 4l + 3 and $1 \leq l \leq ⌊\frac{n}{4}⌋ - 1$ . Since d (u_{k, p+4l+1}, w)≠d (u_{k, p+4l+2}, w) where w ∈ S − W_k and d (u_{k, p+4l+1}, w)≠d (u_{k, p+4l+2}, w) where w ∈ W_k such that r_m (u_{k, p+4l+1}|S)≠r_m (u_{k, p+4l+2}|S). Since d (u_{k, p+4l+2}, w) = d (u_{k, p+4l+3}, w) where w ∈ S − W_k and d (u_{k, p+4l+2}, w)≠d (u_{k, p+4l+3}, w) where w ∈ W_k such that r_m (u_{k, p+4l+2}|S)≠r_m (u_{k, p+4l+3}|S).

Based on the representation above that every two adjacent vertices have distinct representations such that S is local m-resolving set and ${md}_{l} (Amal (F_{n}, v, m)) \leq m . (⌊\frac{p}{4}⌋ + ⌊\frac{n - p}{4}⌋)$ .

Furthermore, we will show that ${md}_{l} (Amal (F_{n}, v, m)) \geq m . (⌊\frac{p}{4}⌋ + ⌊\frac{n - p}{4}⌋)$ . Take any S ⊂ V (Amal (F_n, v, m)) with |P|<|S|. Suppose $| P | = | S | - 1 = m . (⌊\frac{p}{4}⌋ + ⌊\frac{n - p}{4}⌋) - 1$ . There is one copy of F_n, (F_n)_k have $m . (⌊\frac{p}{4}⌋ + ⌊\frac{n - p}{4}⌋) - 1$ . Let u_k,l ∉ P for p + 1 < l ≤ n − 1 or l ≡ p + 4s with $1 \leq s \leq ⌊\frac{n - p}{4}⌋$ and d (u_{k, p+4(s−1)+1}, u_{k, p+4(s+1)−1}) > 4, then $r_{m} (u_{k, p + 4 (s - 1) + 2}| W_{k}) = \dots = r_{m} (u_{k, p + 4 (s + 1) - 2}| W_{k}) = \underset{⌊\frac{n - p}{4}⌋ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, p + 4 (s - 1) + 2}, w) = \dots = d (u_{k, p + 4 (s + 1) - 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, p + 4 (s - 1) + 2}| P) = \dots = r_{m} (u_{k, p + 4 (s + 1) - 2}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m). Thus, ${md}_{l} (Amal (F_{n}, v, m)) = m . (⌊\frac{p}{4}⌋ + ⌊\frac{n - p}{4}⌋)$ .

Sub Case 1.2.

For n − p ≡ 3(mod4)

Choose S = {u_j,r;1 ≤ j ≤ m, r ≡ p (mod4), r > p} ∪ {u_j,n−1} obtained the vertex representation as follows

1. For u_k,l ∈ V (Amal (F_n, v, m)) and 1 ≤ l ≤ p − 1, d (u_k,1, w)≠d (u_k,2, w) where w ∈ S − W_k such that r_m (u_k,1|S − W_k)≠r_m (u_k,2|S − W_k). Since d (u_k,1, w) = d (u_k,2, w) where w ∈ W_k such that r_m (u_k,1|W_k)≠r_m (u_k,2|W_k). Thus, r_m (u_k,1|S)≠r_m (u_k,2|S).
2. For u_k, v ∈ V (Amal (F_n, v, m)), d(v, w)≠d (u_k, w) where w ∈ S − W_k such that r_m(v|S − W_k)≠r_m (u_k|S − W_k). Since d(v, w)≠d (u_k, w) where w ∈ W_k such that r_m(v|W_k)≠r_m (u_k|W_k). Thus, r_m(v|S)≠r_m (u_k|S).
3. For u_k, u_k,l ∈ V ((F_n)_k) where l≠p (mod4) and u_k,l where l≠n − 2 for n − p ≡ 3(mod4). Since u_k adjacent to w where w ∈ W_k, such that r_m (u_k|W_k)≠r_m (u_k,l|W_k). Since d (u_k, w)≠d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k)≠r_m (u_k,l|S − W_k) with l≠1. Since d (u_k, w) = d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k) = r_m (u_k,l|S − W_k) with l = 1. Thus, r_m(v|S)≠r_m (u_k|S).
4. For u_{k, i} ∈ V ((F_n)_k) and p + 1 ≤ i ≤ p + 3. Since d (u_{k, p+1}, w)≠d (u_{k, p+2}, w) where w ∈ S − W_k and d (u_{k, p+1}, w)≠d (u_{k, p+2}, w) where w ∈ W_k such that r_m (u_{k, p+1}|S)≠r_m (u_{k, p+2}|S). Since d (u_{k, p+2}, w) = d (u_{k, p+3}, w) where w ∈ S − W_k and d (u_{k, p+2}, w)≠d (u_{k, p+3}, w) where w ∈ W_k such that r_m (u_{k, p+2}|S)≠r_m (u_{k, p+3}|S).
5. For u_{k, i} ∈ V ((F_n)_k) where p + 4l + 1 ≤ i ≤ p + 4l + 3 and $1 \leq l \leq ⌊\frac{n}{4}⌋ - 1$ . Since d (u_{k, p+4l+1}, w)≠d (u_{k, p+4l+2}, w) where w ∈ S − W_k and d (u_{k, p+4l+1}, w)≠d (u_{k, p+4l+2}, w) where w ∈ W_k such that r_m (u_{k, p+4l+1}|S)≠r_m (u_{k, p+4l+2}|S). Since d (u_{k, p+4l+2}, w) = d (u_{k, p+4l+3}, w) where w ∈ S − W_k and d (u_{k, p+4l+2}, w)≠d (u_{k, p+4l+3}, w) where w ∈ W_k such that r_m (u_{k, p+4l+2}|S)≠r_m (u_{k, p+4l+3}|S).
6. For u_{k, i} ∈ V ((F_n)_k) where i = n − 2, n. Since u_k,n−2 is not adjacent to u_{k, n}. It is clear.

Based on the representation above that every two adjacent vertices has distinct representations such that S is local m-resolving set and ${md}_{l} (Amal (F_{n}, v, m)) \leq m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉)$ .

Furthermore, we will show that ${md}_{l} (Amal (F_{n}, v, m)) \geq m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉)$ . Take any S ⊂ V (Amal (F_n, v, m)) with|P|<|S|. Suppose $| P | = | S | - 1 = m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉) - 1$ . There are one copies of F_n, (F_n)_k have $m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉) - 1$ . There are two possibilities namely 1) u_k,l ∉ P for 1 < l < n − 1 and 2) u_k,n−1 ∉ P.

1. Let u_k,l ∉ P for 1 < l ≤ n − 1 or l ≡ p + 4s with $1 \leq s \leq ⌊\frac{n - p}{4}⌋$ and d (u_{k, p+4(s−1)+1}, u_{k, p+4(s+1)−1}) > 4, then $r_{m} (u_{k, p + 4 (s - 1) + 2}| W_{k}) = \dots = r_{m} (u_{k, p + 4 (s + 1) - 2}| W_{k}) = \underset{⌊\frac{n - p}{4}⌋}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, p + 4 (s - 1) + 2}, w) = \dots = d (u_{k, p + 4 (s + 1) - 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, p + 4 (s - 1) + 2}| P) = \dots = r_{m} (u_{k, p + 4 (s + 1) - 2}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
2. Let u_k,n−1 ∉ P. Since $r_{m} (u_{k, n - 1}| W_{k}) = r_{m} (u_{k, n}| W_{k}) = \underset{⌊\frac{n - p}{4}⌋}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since d (u_k,n−1, w) = d (u_{k, n}, w) for w ∈ P − W_k and r_m (u_k,n−1|P) = r_m (u_{k, n}|P), then P is not local m-resolving set of Amal (F_n, v, m).

Based on 1) and 2) that ${md}_{l} (Amal (F_{n}, v, m)) \geq m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉)$ . Thus, ${md}_{l} (Amal (F_{n}, v, m)) = m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉)$ .

Case 2.

For $4 \leq p \leq ⌈\frac{n}{4}⌉$

Sub Case 2.1.

For n − p ≡ 0, 1, 2(mod4)

We have some conditions for the set S as follows:

a for p ≡ 1(mod4), S = {u_{j, t};1 ≤ j ≤ m, t ≡ 1(mod4)} ∪ {u_j,r;1 ≤ j ≤ m, r ≡ p (mod4), r > p}.
b for p ≡ 2(mod4), S = {u_{j, t};1 ≤ j ≤ m, t ≡ 2(mod4)} ∪ {u_j,r;1 ≤ j ≤ m, r ≡ p (mod4), r > p}.
c for p ≡ 3(mod4), S = {u_{j, t};1 ≤ j ≤ m, t ≡ 3(mod4)} ∪ {u_j,r;1 ≤ j ≤ m, r ≡ p (mod4), r > p}.
d for p ≡ 0(mod4), S = {u_{j, t};1 ≤ j ≤ m, t ≡ 0(mod4)} ∪ {u_j,r;1 ≤ j ≤ m, r ≡ p (mod4), r > p}.

We obtained the vertex representation as follows:

1. For u_{k, i} ∈ V ((F_n)_k) where p − 3 ≤ i ≤ p − 1 and p ≡ 0, 1, 2, 3(mod4). Since d (u_{k, p−1}, w)≠d (u_{k, p−2}, w) where w ∈ S − W_k and d (u_{k, p−1}, w)≠d (u_{k, p−2}, w) where w ∈ W_k such that r_m (u_{k, p−1}|S)≠r_m (u_{k, p−2}|S). Since d (u_{k, p−2}, w) = d (u_{k, p−3}, w) where w ∈ S − W_k and d (u_{k, p−2}, w)≠d (u_{k, p−3}, w) where w ∈ W_k such that r_m (u_{k, p−2}|S)≠r_m (u_{k, p−3}|S).
2. For u_{k, i} ∈ V ((F_n)_k) where p − 4l + 1 ≤ i ≤ p − 4l + 3, $2 \leq l \leq ⌊\frac{n}{4}⌋ - 1$ , and ap ≡ 0, 1, 2, 3(mod4). Since d (u_{k, p−4l+1}, w)≠d (u_{k, p−4l+2}, w) where w ∈ S − W_k and d (u_{k, p−4l+1}, w)≠d (u_{k, p−4l+2}, w) where w ∈ W_k such that r_m (u_{k, p−4l+1}|S)≠r_m (u_{k, p−4l+2}|S). Since d (u_{k, p−4l+2}, w) = d (u_{k, p−4l+3}, w) where w ∈ S − W_k and d (u_{k, p−4l+2}, w)≠d (u_{k, p−4l+3}, w) where w ∈ W_k such that r_m (u_{k, p−4l+2}|S)≠r_m (u_{k, p−4l+3}|S).
3. For u_k,l ∈ V (Amal (F_n, v, m)) and 1 ≤ l ≤ p − 1, d (u_k,1, w)≠d (u_k,2, w) where w ∈ S − W_k such that r_m (u_k,1|S − W_k)≠r_m (u_k,2|S − W_k). Since d (u_k,1, w) = d (u_k,2, w) where w ∈ W_k such that r_m (u_k,1|W_k)≠r_m (u_k,2|W_k). Thus, r_m (u_k,1|S)≠r_m (u_k,2|S).
4. For u_k, v ∈ V (Amal (F_n, v, m)), d(v, w)≠d (u_k, w) where w ∈ S − W_k such that r_m(v|S − W_k)≠r_m (u_k|S − W_k). Since d(v, w)≠d (u_k, w) where w ∈ W_k such that r_m(v|W_k)≠r_m (u_k|W_k). Thus, r_m(v|S)≠r_m (u_k|S).
5. For u_k, u_k,l ∈ V ((F_n)_k) where l≠p (mod4) and u_k,l where l≠n − 2 for n − p ≡ 2(mod4). Since u_k adjacent to w where w ∈ W_k, such that r_m (u_k|W_k)≠r_m (u_k,l|W_k). Since d (u_k, w)≠d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k)≠r_m (u_k,l|S − W_k) with l≠1. Since d (u_k, w) = d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k) = r_m (u_k,l|S − W_k) with l = 1. Thus, r_m(v|S)≠r_m (u_k|S).
6. For u_{k, i} ∈ V ((F_n)_k) and p + 1 ≤ i ≤ p + 3. Since d (u_{k, p+1}, w)≠d (u_{k, p+2}, w) where w ∈ S − W_k and d (u_{k, p+1}, w)≠d (u_{k, p+2}, w) where w ∈ W_k such that r_m (u_{k, p+1}|S)≠r_m (u_{k, p+2}|S). Since d (u_{k, p+2}, w) = d (u_{k, p+3}, w) where w ∈ S − W_k and d (u_{k, p+2}, w)≠d (u_{k, p+3}, w) where w ∈ W_k such that r_m (u_{k, p+2}|S)≠r_m (u_{k, p+3}|S).
7. For u_{k, i} ∈ V ((F_n)_k) where p + 4l + 1 ≤ i ≤ p + 4l + 3 and $1 \leq l \leq ⌊\frac{n}{4}⌋ - 1$ . Since d (u_{k, p+4l+1}, w)≠d (u_{k, p+4l+2}, w) where w ∈ S − W_k and d (u_{k, p+4l+1}, w)≠d (u_{k, p+4l+2}, w) where w ∈ W_k such that r_m (u_{k, p+4l+1}|S)≠r_m (u_{k, p+4l+2}|S). Since d (u_{k, p+4l+2}, w) = d (u_{k, p+4l+3}, w) where w ∈ S − W_k and d (u_{k, p+4l+2}, w)≠d (u_{k, p+4l+3}, w) where w ∈ W_k such that r_m (u_{k, p+4l+2}|S)≠r_m (u_{k, p+4l+3}|S).

Based on the representation above that every two adjacent vertices has distinct representations such that S is a local m-resolving set and ${md}_{l} (Amal (F_{n}, v, m)) \leq m . (⌊\frac{p}{4}⌋ + ⌊\frac{n - p}{4}⌋)$ .

1. Let u_k,l ∉ P for p + 1 < l ≤ n − 1 or l ≡ p + 4s with $1 \leq s \leq ⌊\frac{n - p}{4}⌋$ and d (u_{k, p+4(s−1)+1}, u_{k, p+4(s+1)−1}) > 4, then $r_{m} (u_{k, p + 4 (s - 1) + 2}| W_{k}) = \dots = r_{m} (u_{k, p + 4 (s + 1) - 2}| W_{k}) = \underset{⌊\frac{n - p}{4}⌋ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, p + 4 (s - 1) + 2}, w) = \dots = d (u_{k, p + 4 (s + 1) - 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, p + 4 (s - 1) + 2}| P) = \dots = r_{m} (u_{k, p + 4 (s + 1) - 2}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
2. Let u_k,l ∉ P for 1 < l ≤ p − 1 or l ≡ 1(mod4) and p ≡ 1(mod4), then $r_{m} (u_{k, p - 4 s - 2}| W_{k}) = \dots = r_{m} (u_{k, p - 4 s + 2}| W_{k}) = \underset{⌊\frac{n - p}{4}⌋ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, p - 4 s - 2}, w) = \dots = d (u_{k, p - 4 s + 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, p - 4 s - 2}| P) = \dots = r_{m} (u_{k, p - 4 s + 2}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).

Thus, ${md}_{l} (Amal (F_{n}, v, m)) = m . (⌊\frac{p}{4}⌋ + ⌊\frac{n - p}{4}⌋)$ .

Sub Case 2.2.

For n − p ≡ 3(mod4)

We have some conditions for the set S as follows

a for p ≡ 1(mod4), S = {u_{j, t};1 ≤ j ≤ m, t ≡ 1(mod4)}∪{u_j,r;1 ≤ j ≤ m, r ≡ p (mod4), r > p}∪{u_j,n−1}.
b for p ≡ 2(mod4), S = {u_{j, t};1 ≤ j ≤ m, t ≡ 2(mod4)}∪{u_j,r;1 ≤ j ≤ m, r ≡ p (mod4), r > p}∪{u_j,n−1}.
c for p ≡ 3(mod4), S = {u_{j, t};1 ≤ j ≤ m, t ≡ 3(mod4)}∪{u_j,r;1 ≤ j ≤ m, r ≡ p (mod4), r > p}∪{u_j,n−1}.
d for p ≡ 0(mod4), S = {u_{j, t};1 ≤ j ≤ m, t ≡ 0(mod4)}∪{u_j,r;1 ≤ j ≤ m, r ≡ p (mod4), r > p}∪{u_j,n−1}.

We obtained the vertex representation as follows:

1. For u_{k, i} ∈ V ((F_n)_k) where p − 3 ≤ i ≤ p − 1 and p ≡ 0, 1, 2, 3(mod4). Since d (u_{k, p−1}, w)≠d (u_{k, p−2}, w) where w ∈ S − W_k and d (u_{k, p−1}, w)≠d (u_{k, p−2}, w) where w ∈ W_k such that r_m (u_{k, p−1}|S)≠r_m (u_{k, p−2}|S). Since d (u_{k, p−2}, w) = d (u_{k, p−3}, w) where w ∈ S − W_k and d (u_{k, p−2}, w)≠d (u_{k, p−3}, w) where w ∈ W_k such that r_m (u_{k, p−2}|S)≠r_m (u_{k, p−3}|S).
2. For u_{k, i} ∈ V ((F_n)_k) where p − 4l + 1 ≤ i ≤ p − 4l + 3, $2 \leq l \leq ⌊\frac{n}{4}⌋ - 1$ , and ap ≡ 0, 1, 2, 3(mod4). Since d (u_k,p−4l+1, w)≠d (u_{k, p−4l+2}, w) where w ∈ S − W_k and d (u_{k, p−4l+1}, w)≠d (u_{k, p−4l+2}, w) where w ∈ W_k such that r_m (u_{k, p−4l+1}|S)≠r_m (u_{k, p−4l+2}|S). Since d (u_{k, p−4l+2}, w) = d (u_{k, p−4l+3}, w) where w ∈ S − W_k and d (u_{k, p−4l+2},w)≠d (u_{k, p−4l+3}, w) where w ∈ W_k such that r_m (u_{k, p−4l+2}|S)≠r_m (u_{k, p−4l+3}|S).
3. For u_k,l ∈ V (Amal (F_n, v, m)) and 1 ≤ l ≤ p − 1, d (u_k,1, w)≠d (u_k,2, w) where w ∈ S − W_k such that r_m (u_k,1|S − W_k)≠r_m (u_k,2|S − W_k). Since d (u_k,1, w) = d (u_k,2, w) where w ∈ W_k such that r_m (u_k,1|W_k)≠r_m (u_k,2|W_k). Thus, r_m (u_k,1|S)≠r_m (u_k,2|S).
4. For u_k, v ∈ V (Amal (F_n, v, m)), d(v, w)≠d (u_k, w) where w ∈ S − W_k such that r_m(v|S − W_k)≠r_m (u_k|S − W_k). Since d(v, w)≠d (u_k, w) where w ∈ W_k such that r_m(v|W_k)≠r_m (u_k|W_k). Thus, r_m(v|S)≠r_m (u_k|S).
5. For u_k, u_k,l ∈ V ((F_n)_k) where l≠p (mod4) and u_k,l where l≠n − 2 for n − p ≡ 3(mod4). Since u_k adjacent to w where w ∈ W_k, such that r_m (u_k|W_k)≠r_m (u_k,l|W_k). Since d (u_k, w)≠d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k)≠r_m (u_k,l|S − W_k) with l≠1. Since d (u_k, w) = d (u_k,l, w) where w ∈ S − W_k such that r_m (u_k|S − W_k) = r_m (u_k,l|S − W_k) with l = 1. Thus, r_m(v|S)≠r_m (u_k|S).
6. For u_{k, i} ∈ V ((F_n)_k) and p + 1 ≤ i ≤ p + 3. Since d (u_{k, p+1}, w)≠d (u_{k, p+2}, w) where w ∈ S − W_k and d (u_{k, p+1}, w)≠d (u_{k, p+2}, w) where w ∈ W_k such that r_m (u_{k, p+1}|S)≠r_m (u_{k, p+2}|S). Since d (u_{k, p+2}, w) = d (u_{k, p+3}, w) where w ∈ S − W_k and d (u_{k, p+2}, w)≠d (u_{k, p+3}, w) where w ∈ W_k such that r_m (u_{k, p+2}|S)≠r_m (u_{k, p+3}|S).
7. For u_{k, i} ∈ V ((F_n)_k) where p + 4l + 1 ≤ i ≤ p + 4l + 3 and $1 \leq l \leq ⌊\frac{n}{4}⌋ - 1$ . Since d (u_{k, p+4l+1}, w)≠d (u_{k, p+4l+2}, w) where w ∈ S − W_k and d (u_{k, p+4l+1}, w)≠d (u_{k, p+4l+2}, w) where w ∈ W_k such that r_m (u_{k, p+4l+1}|S)≠r_m (u_{k, p+4l+2}|S). Since d (u_{k, p+4l+2}, w) = d (u_{k, p+4l+3}, w) where w ∈ S − W_k and d (u_{k, p+4l+2}, w)≠d (u_{k, p+4l+3}, w) where w ∈ W_k such that r_m (u_{k, p+4l+2}|S)≠r_m (u_{k, p+4l+3}|S).
8. For u_{k, i} ∈ V ((F_n)_k) where i = n − 2, n. Since u_k,n−2 does not adjacent to u_{k, n}. It is clear.

Based on the representation above that every two adjacent vertices have distinct representations such that S is a local m-resolving set and ${md}_{l} (Amal (F_{n}, v, m)) \leq m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉)$ .

Furthermore, we will show that ${md}_{l} (Amal (F_{n}, v, m)) \geq m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉)$ . Take any S ⊂ V (Amal (F_n, v, m)) with|P|<|S|. Suppose $| P | = | S | - 1 = m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉) - 1$ . There are one copies of F_n, (F_n)_k have $m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉) - 1$ . There are three possibilities namely 1) u_k,l ∉ P for 1 < l < n − 1 and 2) u_k,n−1 ∉ P.

1. Let u_k,l ∉ P for 1 < l ≤ n − 1 or l ≡ p + 4s with $1 \leq s \leq ⌊\frac{n - p}{4}⌋$ and d (u_{k, p+4(s−1)+1}, u_{k, p+4(s+1)−1}) > 4, then $r_{m} (u_{k, p + 4 (s - 1) + 2}| W_{k}) = \dots = r_{m} (u_{k, p + 4 (s + 1) - 2}| W_{k}) = \underset{⌊\frac{n - p}{4}⌋}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, p + 4 (s - 1) + 2}, w) = \dots = d (u_{k, p + 4 (s + 1) - 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, p + 4 (s - 1) + 2}| P) = \dots = r_{m} (u_{k, p + 4 (s + 1) - 2}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).
2. Let u_k,n−1 ∉ P. Since $r_{m} (u_{k, n - 1}| W_{k}) = r_{m} (u_{k, n}| W_{k}) = \underset{⌊\frac{n - p}{4}⌋}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since d (u_k,n−1, w) = d (u_{k, n}, w) for w ∈ P − W_k and r_m (u_k,n−1|P) = r_m (u_{k, n}|P), then P is not local m-resolving set of Amal (F_n, v, m).
3. Let u_k,l ∉ P for 1 < l ≤ p − 1 or l ≡ 1(mod4) and p ≡ 1(mod4), then $r_{m} (u_{k, p - 4 s - 2}| W_{k}) = \dots = r_{m} (u_{k, p - 4 s + 2}| W_{k}) = \underset{⌊\frac{n - p}{4}⌋ - 1}{\underset{⏟}{\{2, 2, 2, \dots, 2\}}}$ . Since $d (u_{k, p - 4 s - 2}, w) = \dots = d (u_{k, p - 4 s + 2}, w)$ for w ∈ P − W_k and $r_{m} (u_{k, p - 4 s - 2}| P) = \dots = r_{m} (u_{k, p - 4 s + 2}| P)$ , then P is not local m-resolving set of Amal (F_n, v, m).

Based on 1), 2) and 3) that ${md}_{l} (Amal (F_{n}, v, m)) \geq m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉)$ . Thus, ${md}_{l} (Amal (F_{n}, v, m)) = m . (⌊\frac{p}{4}⌋ + ⌈\frac{n - p}{4}⌉)$ . □

Conclusion

We have characterized the local multiset dimension of amalgamation graphs. We have found the upper bound of local multiset dimension and determined the exact value of local multiset dimension of path P_n, complete graph K_n, wheel graph W_n, and fan graph F_n. There are some graphs that attain the upper bound of local multiset dimension namely wheel graphs. On the otherhand, we found the following problem, as follows.

Open Problem 0.1 Determine the lower bound of local multiset dimension of amalgamation graphs.

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Acknowledgments

We gratefully acknowledge the support from Universitas Airlangga and University of Jember of year 2022.

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7. Alfarisi R, Lin Y, Ryan J, et al.: A Note on Multiset Dimension and Local Multiset Dimension of Graphs. Stat. Optim. Inf. Comput. 2019; 8: 890–901. Accepted. Publisher Full Text
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10. Iswadi H, Baskoro ET, Salman ANM, et al.: The resolving graph of amalgamation of cycles Util. Math. 2010; 83: 121–132.
11. Diestel R: Graph Theory, Graduate Texts in Mathematics. Springer; 2016. 978-3-642-14278-91.
12. Alfarisi R, SusilowatiDafik L: The Local Multiset Resolving of Graphs.2022. In Review.

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Author details Author details

Ridho Alfarisi
Roles: Conceptualization, Investigation, Methodology, Resources, Writing – Original Draft Preparation, Writing – Review & Editing

Liliek Susilowati
Roles: Writing – Review & Editing

Dafik Dafik
Roles: Writing – Review & Editing

Savari Prabhu
Roles: Resources, Validation, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

This work was supported in part by University of Jember and Universitas Airlangga, Indonesia.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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Alfarisi R, Susilowati L, Dafik D and Prabhu S. Local Multiset Dimension of Amalgamation Graphs [version 2; peer review: 2 approved]. F1000Research 2024, 12:95 (https://doi.org/10.12688/f1000research.128866.2)

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Reviewer Report 03 Jun 2024

Muhammad Kamran Siddique, Department of Mathematics, COMSATS University Islamabad, Lahore, Pakistan

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Reviewer Report 31 Oct 2023

Muhammad Kamran Siddique, Department of Mathematics, COMSATS University Islamabad, Lahore, Pakistan

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https://doi.org/10.5256/f1000research.141499.r161414

Local Multiset Dimension of Amalgamation Graphs

I found that all the mathematical work is correct. Moreover paper is clearly written in a good way. It is new, as well as good contribution in the current area of ... Continue reading

The definitions of m-resolving set and local m-resolving set conveys the same meaning. The condition xy=E(G), should be removed from the definition of m-resolving set.
On page no. 4, local multiset dimension is denoted by μ_l(G). But md_l(G) is used in definition and other results.
In the illustration of local m-resolving set, W={v₁} is the considered local m-resolving set, but the vertex representations are given with respect to Π.
In figure 2, the terminal vertex v₀ is not labelled in the graphs.
In the line above Lemma 0.2, it is mentioned that upper bound for Amal(G,v,m) is given.
In the proof of Lemma 0.2, the terminal vertex v is not included in the vertex set of Amal(G,v,m). Instead, a vertex u is mentioned. The same mistake can be seen in Theorem 0.3 and Theorem 0.4.
On page no. 6, in the proof of Lemma 0.2, it is given that u_k,r_, u_k,s∈V(G) are at equal distance from every w∈S-W_k and it is concluded that their representations differ with respect to W.
In the edge set of Amal(P_n,v,m) [Theorem 0.3] and Amal(K_n,v,m) [Theorem 0.4], an extra curly brace is placed.
In the proof of Theorem 0.4, two adjacent vertices x and y are mentioned to be equidistant from every w∈V(Amal(K_n,v,m)). Followed by that, it is given that they have different representation w.r.t W.
In the edge set of Amal(W_n,v,m), j=1mE(Wnj) is typed as j=1 j=mE(Wnj). Similarly, for F_n
The set P is not mentioned clearly in proof of Theorem 0.5, Theorem 0.7, Theorem 0.8 and Theorem 0.9.
In the proof of Theorem 0.7, (page no. 10, first line) ‘local m-resolvings of F_n’ should be changed as ‘local m-resolving set of F_n’.
It is mentioned in many places that there are three possibilities but followed by that only two cases are given.
Cite following recent paper about this topic¹^-⁶.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Partly
Are the conclusions drawn adequately supported by the results?

Yes

References

1. Zhang X, Wu X, Akhter S, Jamil M, et al.: Edge-Version Atom-Bond Connectivity and Geometric Arithmetic Indices of Generalized Bridge Molecular Graphs. Symmetry. 2018; 10 (12). Publisher Full Text
2. Zhang X, Awais H, Javaid M, Siddiqui M: Multiplicative Zagreb Indices of Molecular Graphs. Journal of Chemistry. 2019; 2019: 1-19 Publisher Full Text
3. Zhang X, Rauf A, Ishtiaq M, Siddiqui M, et al.: On Degree Based Topological Properties of Two Carbon Nanotubes. Polycyclic Aromatic Compounds. 2022; 42 (3): 866-884 Publisher Full Text
4. Zhang X, Jiang H, Liu JB, Shao Z: The Cartesian Product and Join Graphs on Edge-Version Atom-Bond Connectivity and Geometric Arithmetic Indices.Molecules. 2018; 23 (7). PubMed Abstract | Publisher Full Text
5. Zhang X, Naeem M, Baig A, Zahid M: Study of Hardness of Superhard Crystals by Topological Indices. Journal of Chemistry. 2021; 2021: 1-10 Publisher Full Text
6. Zhang X, Siddiqui MK, Javed S, Sherin L, et al.: Physical Analysis of Heat for Formation and Entropy of Ceria Oxide Using Topological Indices.Comb Chem High Throughput Screen. 2022; 25 (3): 441-450 PubMed Abstract | Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Graph theory. Discrete Mathematics, Chemical graph Theory, Combinatorics

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

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Liliek Susilowati, Mathematics, Universitas Airlangga, Surabaya, 68121, Indonesia

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Thank you for your suggestion.
I will revise my paper accordingly to your comments and suggestions.
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Thank you for your suggestion.
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Reviewer Report 26 Jan 2023

Ismail Naci Cangul, Department of Mathematics, Bursa Uludag University, Gorukle, Turkey

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https://doi.org/10.5256/f1000research.141499.r161413

Set dimension in graph theory is a very applicable area which collected a lot of attention by many people. Here, the authors continued their research with local multiset dimension as a variant and studied this number for amalgamation graphs. Amalgamation is one of the graph operations and this study can easily be carried to other graph operations. I also wondered that this ideas can be applied to algebra as amalgamation plays an important role in algebraic structures.

This manuscript is written very clearly as a new step in continuing theory and it will add value to the area.

The abstract seems to be uncompiled as there are "$" signs all around. Also Theorems, Propositions, Lemmas must be written in bold to get the attention of the reader.

Very nice work on multiset theory.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Graph theory, number theory, discrete group theory, Molecular graphs

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Liliek Susilowati, Mathematics, Universitas Airlangga, Surabaya, 68121, Indonesia

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Dear Prof Ismail,

For the abstract as there are "$" signs all around. The symbol "$", only editors can fix or remove the "$" sign. I've reported to the ... Continue reading Dear Prof Ismail,

For the abstract as there are "$" signs all around. The symbol "$", only editors can fix or remove the "$" sign. I've reported to the editor to remove the "$" sign.

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Dr. Liliek Susilowati and Dr. Ridho Alfarisi
Dear Prof Ismail,

For the abstract as there are "$" signs all around. The symbol "$", only editors can fix or remove the "$" sign. I've reported to the editor to remove the "$" sign.

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Liliek Susilowati, Mathematics, Universitas Airlangga, Surabaya, 68121, Indonesia

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Dear Prof Ismail,

For the abstract as there are "$" signs all around. The symbol "$", only editors can fix or remove the "$" sign. I've reported to the ... Continue reading Dear Prof Ismail,

For the abstract as there are "$" signs all around. The symbol "$", only editors can fix or remove the "$" sign. I've reported to the editor to remove the "$" sign.

Best Regards
Dr. Liliek Susilowati and Dr. Ridho Alfarisi
Dear Prof Ismail,

For the abstract as there are "$" signs all around. The symbol "$", only editors can fix or remove the "$" sign. I've reported to the editor to remove the "$" sign.

Best Regards
Dr. Liliek Susilowati and Dr. Ridho Alfarisi
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Ismail Naci Cangul, Bursa Uludag University, Gorukle, Turkey
Muhammad Kamran Siddique, COMSATS University Islamabad, Lahore, Pakistan

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03 Jun 2024 | for Version 2

Muhammad Kamran Siddique, Department of Mathematics, COMSATS University Islamabad, Lahore, Pakistan

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Approved

The authors have carefully addressed all the reviewing comments raised in the first round of review. It's my pleasure to recommend acceptance of the current version.

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No competing interests were disclosed.

Reviewer Expertise

Graph theory. Discrete Mathematics, Chemical graph Theory, Combinatorics

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31 Oct 2023 | for Version 1

Muhammad Kamran Siddique, Department of Mathematics, COMSATS University Islamabad, Lahore, Pakistan

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Approved With Reservations

The definitions of m-resolving set and local m-resolving set conveys the same meaning. The condition xy=E(G), should be removed from the definition of m-resolving set.
On page no. 4, local multiset dimension is denoted by μ_l(G). But md_l(G) is used in definition and other results.
In the illustration of local m-resolving set, W={v₁} is the considered local m-resolving set, but the vertex representations are given with respect to Π.
In figure 2, the terminal vertex v₀ is not labelled in the graphs.
In the line above Lemma 0.2, it is mentioned that upper bound for Amal(G,v,m) is given.
In the proof of Lemma 0.2, the terminal vertex v is not included in the vertex set of Amal(G,v,m). Instead, a vertex u is mentioned. The same mistake can be seen in Theorem 0.3 and Theorem 0.4.
On page no. 6, in the proof of Lemma 0.2, it is given that u_k,r_, u_k,s∈V(G) are at equal distance from every w∈S-W_k and it is concluded that their representations differ with respect to W.
In the edge set of Amal(P_n,v,m) [Theorem 0.3] and Amal(K_n,v,m) [Theorem 0.4], an extra curly brace is placed.
In the proof of Theorem 0.4, two adjacent vertices x and y are mentioned to be equidistant from every w∈V(Amal(K_n,v,m)). Followed by that, it is given that they have different representation w.r.t W.
In the edge set of Amal(W_n,v,m), j=1mE(Wnj) is typed as j=1 j=mE(Wnj). Similarly, for F_n
The set P is not mentioned clearly in proof of Theorem 0.5, Theorem 0.7, Theorem 0.8 and Theorem 0.9.
In the proof of Theorem 0.7, (page no. 10, first line) ‘local m-resolvings of F_n’ should be changed as ‘local m-resolving set of F_n’.
It is mentioned in many places that there are three possibilities but followed by that only two cases are given.
Cite following recent paper about this topic¹^-⁶.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Partly
Are the conclusions drawn adequately supported by the results?

Yes

References

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Graph theory. Discrete Mathematics, Chemical graph Theory, Combinatorics

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Reviewer Report

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26 Jan 2023 | for Version 1

Ismail Naci Cangul, Department of Mathematics, Bursa Uludag University, Gorukle, Turkey

16 Views Cite this report Responses(1)

Approved

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Graph theory, number theory, discrete group theory, Molecular graphs

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Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

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Local Multiset Dimension of Amalgamation Graphs

Abstract

Keywords

Revised Amendments from Version 1

Introduction

Figure 1. (a) A graph with multiset dimension 3; (b) A graph with local multiset dimension 1.

Lemma 0.1

Proposition 0.1

Theorem 0.1

Theorem 0.2

Proposition 0.2

Definition 0.1

Figure 2. a) Amal(Hj, v) with j = 1, 2, 3 and b) Amal(H1, v, 4).

Methods

Figure 3. Flowchart of the extrapolate design of the study.

Results

Lemma 0.2

Proof.

Theorem 0.3

Proof.

Theorem 0.4

Proof.

Theorem 0.5

Proof.

Theorem 0.6

Proof.

Case 1.

Case 2.

Case 3.

Case 4.

Theorem 0.7

Proof.

Theorem 0.8

Proof.

Theorem 0.9

Proof.

Case 1.

Sub Case 1.1.

Sub Case 1.2.

Case 2.

Sub Case 2.1.

Sub Case 2.2.

Conclusion

Data availability

Acknowledgments

References

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Figure 2. a) Amal(H_j, v) with j = 1, 2, 3 and b) Amal(H₁, v, 4).