key: cord-0973177-ysobx4gx
authors: Bera, Sanchari; Pal, Madhumangal
title: A novel concept of domination in m-polar interval-valued fuzzy graph and its application
date: 2021-08-29
journal: Neural Comput Appl
DOI: 10.1007/s00521-021-06405-9
sha: 7f041abc1f15a19204ba25982764df385176741f
doc_id: 973177
cord_uid: ysobx4gx

The concept of domination is one of the most significant topics in graph theory to handle unpredictable phenomena. In this study, an unprecedented idea of domination is introduced in m-polar interval-valued fuzzy graph (m-PIVFG). Domination number (DN), isolated vertex, total dominating set, independent set of domination on m-PIVFG are discussed. Some algebraic properties of domination on m-PIVFG are investigated. Weak domination, strong domination, split and non-split domination, cototal and global dominating sets on m-PIVFG are investigated with some fundamental hypotheses and models. We explore the concept of domination in m-PIVFG by solving a case study of locating new facilities to handle a catastrophe reaction activity due to the “COVID-19 pandemic” in West Bengal, India. Ultimately, conclusions and avenues of future scopes are placed at the end of this study.

Fuzzy set FG Fuzzy graph DS Dominating set DN Domination number IVFS Interval-valued fuzzy set IVFG Interval-valued fuzzy graph m-PFS m-polar fuzzy set m-PFG m-polar fuzzy graph m-PIVFS m-polar interval-valued fuzzy set m-PIVFG m-polar interval-valued fuzzy graph

Initially, graph theory plays an important role in various fields of mathematics as well as computer science such as group theory, topology, diagram representation, probability, numerical analysis, matrix theory and so on. Later day by day it has spread its branches in various fields of physics, biochemistry, biology, engineering field, operations research, astronomy, management science and most importantly in computer science. Graphs are the pictorial portrayal that connects the objects and features their data. To signify a real-world issue, this connection is most important. The fuzzy concept deals with haziness in operation research, human thinking, pattern recognition, expert systems, computer science, management science, control engineering, artificial intelligence, robotics, clustering analysis, and so on. Watching the immense utilization of graph theory, a numerical structure characterized as G ¼ ðV; EÞ along with a great deal of vertices V and edges E was introduced. When there is a case of uncertainty and ambiguity in connection in either vertices or edges or both then the corresponding graph can be transferred to the fuzzy graph.

Fuzzy graph theory has wide range of applications in decision-making problems, facility location problems, covering problems. The fuzzy graph theory is tracking down an expanding number of uses for demonstrating continuous frameworks where the degree of data innate in the framework relies upon various levels of exactness. These days, scientists have two primary issues. From one perspective, people depend upon basic frameworks like a few basic infections, transportation, power, and sewage systems, water supply. Fuzzy graph can be used in different facility problems which are listed following:

• Health issue (Hospitals, ambulance) • Electricity transmission and distribution • Oil, gas and minerals production, transport and distribution • Networking systems • Water supply and drainage system • Financial areas (Post office, Bank) • Transport systems (Bus, rail, airports, ship).

Second issue is discovering optimal solution. It is found in numerous spaces of current science, innovation and financial aspects. After observing these two types of problems, location routing problems have been introduced in this paper. The world is now facing most dangerous COVID-19 pandemic situation. As, the decision making is the main piece of our life, which has a markable importance during this COVID-19 pandemic and it will profoundly influence our day by day life; so this paper is of deep significance to examine and make fundamental strategies to beat the present circumstance.

The classical set theory introduced by Cantor cannot be dealing with this haziness and ambiguity. In 1736, Euler displayed the concept about Graph from the Konigsberg bridge problem. For taking care of different dynamic issues in real life, fuzzy set along with fuzzy relation have been applied. Based on Zadeh's conception on the fuzzy set, Kauffman [20] initiated the idea about the fuzzy graph. Later on, Rosenfeld [31] discussed this concept and developed definitions of various fuzzy notions like fuzzy vertex and edges, fuzzy paths, fuzzy cycles and so on which have been successfully implemented in medical diagnosis, engineering and manufacturing science etc. Afterwards, Bhattacharya [8] incorporated a few comments on fuzzy graphs in 1987. The opinion of a fuzzy set and bipolar fuzzy sets were extended by Zhang [41, 42] in 1994. Mordeson and Peng [23] explored the idea about complement on fuzzy graph and shown a couple of properties and operations on it. Shanon and Atnassov [34] incorporated the idea of intuitionistic fuzzy graphs and some properties on it. Afterwards, this concept was depicted by Sunitha and Kumar [37, 38] . Nagoorgani and Radha [26] presented a regular fuzzy graph to discover the density of fuzzy graph in 2008. Nagoorgani and Malarvizhu [25] studied on isomorphic properties and defined self-complementary fuzzy graphs. A deeper study on IVFG was done by Hongmei and Lianhua [18] in 2009 and by Akram and Dudek [1] in 2011. The ideas of bipolar fuzzy set and m-PFS were detailed by Chen et al. [9] in 2014. A couple of properties identified with isomorphism and supplement of IVFG were focused by Talebi and Rashmanlou [39] . Afterwards, Ghorai and Pal [14] [15] [16] [17] examined several belongings on m-PFGs such as isomorphism, density and some other operations on it. Pramanik and pal [29] investigated different kinds of planar graphs. In fact, various extensions of fuzzy graphs are incorporated to handle the dubiousness of this present reality issue. Fuzzy graphs are used several models to tackle different dynamic issues in this uncertain environment. Various sorts of examination on summed up fuzzy graphs were handled by Samanta and Pal [32, 33] . Recently in 2019, radio fuzzy graphs were discussed by Mahapatra et al. [21] . Most recently in 2020, Jana et al. [19] studied on bipolar fuzzy dombi prioritized aggregation operators.

During the 1850s, an investigation of DSs in graph theory began simply as an issue in the round of chess. Chess devotees in Europe observed the issue of deciding the minimum number of queens that can be put on a chessboard with the goal that all the squares are either assaulted by a queen or involved by a queen. In 1962, the idea of domination in graphs was presented by Ore [27] and Berge [7] , and thereafter concentrated by Cockayne and Hedetniemi [10, 11] . Utilizing effective edges Somasundaram [35, 36] explored domination in fuzzy graphs. Afterwards, utilizing strong arc segments domination in the fuzzy graph was talked about by Nagoorgani and Chandrasekaran [24] . In the fuzzy graph, the concept of strong and weak domination was initiated by Gani and Ahamed [13] . The idea of 2-DS and secure DSs were investigated by Merouane and Chellali [22] in 2015.

Motivating from different types of research on domination in fuzzy graphs [28] , we focus on exploring domination in m-PIVFG. A more detailed classification of some related research is presented in Table 1 .

The primary commitment of this investigation is as per the following:

• A novel conceptualization of domination on m-PIVFGs is introduced. 

The remainder of the paper is sorted out as follows: The first section describes the historical backgrounds of graph theory. Our point and inspiration are additionally noted down in Section 1. Some preliminary basic ideas regarding 

In this piece of this paper, we quickly concentrate some essential phrasings of FS, FGs, m-PFS and m-PFGs which are utilized here. The essential meaning of IFGs is additionally illustrated.

In 1965, fuzzy sets, augmentation of the old style idea of the set were suggested by Zadeh [40] . We recall the definitions of m-PFG and IVFG from [5] . For more clarification, one can visit that manuscript thoroughly. All through this article, we utilize the documentation G Ã as an old style classical graph and G ¼ ðV; A; BÞ as an m-PIVFG. Neural Computing and Applications l l B ðxyÞ; and p i l u B ðxyÞ ¼ minfp i l u A ðxÞ; p i l u A ðyÞg À p i l u B ðxyÞ for every i and x; y 2 V: Definition 3 [36] A dominating set S in G ¼ ðV; EÞ is defined as for every x 2 V À S dominates some y 2 S; where S V:

In the next section, domination on m-PIVFG is defined with suitable examples. Some basic terminologies related to domination are also explained.

Herein, the concept of domination in m-PIVFG is explored with examples. In this section, various terms related to domination on m-PIVFG are also discussed. Also, in this part we described minimal domination, maximal domination and total domination in m-PIVFG with suitable examples. Some significant outcomes are likewise clarified with their proofs. 

; 8x 2 V and 8xy 2 E: Definition 8 LetS V: Then cardinality ofS; 

The domination number Ç ðGÞ means the minimum cardinality of all DS in G and i. NðxÞ \D ¼ / ii. There is a vertex y 2 V ÀD such that NðyÞ \D ¼ x;

for each x 2D:

Proof For a minimal DSD of an m-PIVFG G ¼ ðV; A; BÞ; for every vertex x 2D; no proper subset ofD that isD 1 1 D À x is DS. Therefore, any vertex inD 1 cannot dominate any vertex y 2 V ÀD 1 : If y ¼ x; then x cannot dominate any vertex inD 1 : If y 6 ¼ x; then y is not dominated byD 1 but is dominated byD: Thus y can dominate only x inD that means NðyÞ \D ¼ x:

Conversely, letD holds one of the two given criterias. If D is not a minimal DS, then 9 x 2D so thatD 1 ¼D À fxg is dominating. That means atleast one vertex inD 1 can dominate x hence (i) does not hold. Again ifD 1 is a DS, then atleast one vertex inD 1 can dominate every vertex in V ÀD that implies (ii) also does not hold, which contradict our assumption. )D must be a minimal DS. Definition 18 For an m-PIVFG G ¼ ðV; A; BÞ; the lower and upper independence number mean the minimum cardinality and maximum cardinality among all maximal independent sets, respectively, and represented byĩðGÞ and IðGÞ; respectively.

The upper independence number of an m-PIVFG G ¼ ðV; A; BÞ denoted by is defined as the maximum cardinality among all maximal independent sets of G.

Example 5 Let us consider G ¼ ðV; A; BÞ (Fig. 3) , where 

Proof let us consider a maximal independent set I. Then from the definition, for every x 2 V À I; the set I [ fxg is not independent. Hence, for every vertex x 2 V À I; 9y 2 I so that y can dominate x which implies I is a DS. ) I is both independent and dominating. Conversely if we suppose that I is not maximal independent set then 9 x 2 V À I such that I [ fxg is independent. Thus x cannot dominate an y in I. That implies, I is not a DS, which contradict our assumption. ) I is a maximal independent set. h Example 8 From (Fig. 4) , Example 7, we consider a DS fa; cg: Thus hV ÀDi ¼ fb; dg is disconnected. Therefore, fa; cg is split DS on this 3-PIVFG G ¼ ðV; A; BÞ:

Corollary 2 Split DS exists if the graph is not complete and either complete contains a non-complete component or at least two non-trivial components.

Corollary 3 Every split DS in an m -PIVFG is DS, i.e., Ç ðGÞ 6 Ç sp ðGÞ:

Theorem 6 A split DSD of an m -PIVFG G is minimal for each vertex x 2D; one of the accompanying holds:

i. 9y 2 V ÀD so that NðyÞ \D ¼ fxg:

ii. x 2D be isolated.

iii. hV ÀD [ fxgi being connected.

Proof Suppose for an minimal split DSD of an m-PIVFG 9x 2D so that X doesn't satisfy any of the above-mentioned conditions. Thereafter from i. & ii.,D 1 ¼D À fxg be DS. Again from the condition iii. hV ÀDi being disconnected impliesD 1 as a split DS, which is a contradiction. h Definition 25 A strong split DSD of G ¼ ðV; A; BÞ is a DS where the induced subgraph hV ÀDi is totally disconnected with at least two vertices. The split DN Ç ssp means the minimum cardinality of all strong split DS.

Example 9 From Example 7, (Fig. 4) we consider a DS fa; cg where the induced subgraph hV ÀDi ¼ fb; dg being disconnected with two vertices. Therefore, fa; cg is strong split DS on this 3-PIVFG.

Theorem 7 A DSD of G ¼ ðV; A; BÞ be strong split DS iff the followings are satisfied.

i. V ÀD has at least two vertices. ii. For any x; y 2 V ÀD; every x À y path consists at least a vertex ofD:

Proof Proof of this theorem is straight forward. h Theorem 8 A strong split DS of G ¼ ðV; A; BÞ be minimal iff for each vertex v 2D; any of the accompanying condition is fulfilled.

i. 9x 2 V ÀD so that x is adjacent to y 2D: ii. y 2D be an isolated vertex.

Definition 26 A DSD of G ¼ ðV; A; BÞ is said to be nonsplit DS if the induced subgraph hV ÀDi is connected.

The non-split DN Ç ns ðGÞ of an m-PIVFG G ¼ ðV; A; BÞ be minimum cardinality of all non-split DS. Since every non-split DS of an m-PIVFG being DS of that m-PIVFG, then Ç ðGÞ Ç ns ðGÞ:

Example 10 From (Fig. 1 ) Example 1, the DS fbg be nonsplit. hV À fbgi ¼ fa; c; d; eg being connected implies Ç ns ðGÞ ¼ 0:9: In this case Ç ðGÞ ¼ Ç ns ðGÞ: And also if we take fa; b; cg as DS then hV À fa; cgi ¼ fb; d; eg is connected & Ç ns ðGÞ ¼ 1:85: That imply, Ç ðGÞ\Ç ns ðGÞ: )Ç ðGÞ Ç ns ðGÞ:

Theorem 9 A non-split DSD of an m -PIVFG G is minimal iff for each vertex y 2D one of the following is satisfied.

i. 9x 2 V ÀD so that NðxÞ \D ¼ fyg ii. y being isolated vertex iii. NðxÞ \ hV ÀDi ¼ h/i: Proof LetD be minimal DS of an m-PIVFG G ¼ ðV; A; BÞ: Suppose there exist a vertex y 2D so that y satisfy none of the above three conditions. Then by theo-remD 1 ¼D À fvg be a DS. By third condition hV ÀD 1 i is connected. This indicateD 1 is a non-split DS of G ¼ ðV; A; BÞ; a contradiction. Hence, our assumption was wrong.

h

Definition 27 A DSD of G ¼ ðV; A; BÞ is a strong nonsplit DS if the induced subgraph hV ÀDi is complete. The Example 11 Lets consider Example 1 (Fig. 1) . Here, if we consider the DS fa; cg then the set fb; d; eg is a cycle. Henceforth, it is a cycle DS on this 3-PIVFG.

Definition 29 A DSD of a connected m-PIVFG is a path non-split DS if the induced subgraph hV ÀDi is a path. The path non-split DN Ç pns is the minimum cardinality of path non-split DS.

Example 12 Lets consider Example 1 (Fig. 1) . Here, if we consider the DSs fa; dg; fbg; fd; eg; fc; eg; ; fb; a; egfb; c; dg then the sets are path non-split DS. 

Domination in m-PIVFG can be applied in a wide variety of several practical problems such as emergency response operations, logistics systems, covering problems, and location-routing problems. Herein, a case study of the domination in m-PIVFG is incorporated in order to validate our proposed concept.

Disasters are remarkable circumstances that require noteworthy logistical management to distribute essential humanitarian products so as to help and give alleviation to casualties. A productive response assists with diminishing social, financial, and ecological effects. In recent times, Such type of circumstances can be only handled by providing essential services such as medical aid, feeding operation, sanitation services, distribution of humanitarian items, etc. So as to give these products and ventures to the influenced locales, and to have the option to rapidly react if there should arise an occurrence of crises, an effective and adaptable flexible framework is necessary. In this regard, the covering problem plays a vital role to manage such kind of situations. The affected regions are addressed as the demand points and considered to be covered only if at least a facility is accessible to supply all the necessary services within a given distance. The main aim of the problem is maximizing the coverage demand points with the minimum number of facilities. Now, due to the ''COVID-19 pandemic,'' Department Of Health and Family Welfare, West Bengal, India is announced that the whole state is divided into 04 Red Zones (see Fig. 5 , marked by red color) which means the regions are mostly affected. The entry and exit of individuals will be seriously limited. Movement of people will only be allowed for the supply of essential goods and services. Nearly everything will stay shut. In these circumstances, the government wishes to locate some distribution centers (DC) so that each Red Zone will be covered with maximizing the effecting region, minimum logistics cost and conveyance time. In fact, the government aims to set up the minimum number of DC to reduce the opening cost of DC. This scenario can be prevented by a 3-PIVFG, G ¼ ðT; RÞ; where T is the 3-polar interval-valued fuzzy vertex set of Red Zones (Table 2 ) and R is the 3-polar interval-valued fuzzy edge set of the relation between zones depending on these criteria such as the effecting region, logistics cost and conveyance time. These criteria are taken as pole, respectively, in the form of some interval taken from [0, 1]. Here, we have taken 10 number of DC fa; b; c; d; e; f ; g; h; i; jg in which we have to find minimum number of DC that means minimal domination (Fig. 6) . Here p(i) indicates the membership value of ith pole. By direct calculation, we find the minimal DS as fb; g; hg: fb; f ; hg and fb; g; jg are also minimal DS. But finding the DN, we observed that fb; f ; hg is the minimum DS and the DN Ç ðGÞ ¼ 2:25: Hence b, f and h are, respectively, taken DC so that all the criteria fulfilled by the Govt.

The steps are given for determination of the best potential facilities in our case study, which are as per the following:

Step 1 Firstly, the vertex set T of V and the edge set R of E are chosen in our m-PIVFG.

Step 2 Then, the m-PIVFG, G ¼ ðT; RÞ is calculated. Now, we find the vertices x j like p i l l B ðx j x k Þ ¼ p i Fig. 6 Pictorial diagram of the application Neural Computing and Applications minfl l A ðx j Þ; l l A ðx k Þg; p i l u B ðx j x k Þ ¼ p i minfl u A ðx j Þ; l u A ðx k Þg; for j 6 ¼ k:

Step 3 The setT j Y of vertices x j is formulated. If [ k x k ¼ Y ÀT j thenT j is a DS, else it's not a DS.

Step 4 Afterwards, we repeat the same steps (Steps 2-3) to seek all the DSsT j of V.

Step 5 Thus, we get all the DSsT j of Y; Stop.

This article has been incorporated the hypothesis of domination with the idea of m-PIVFG and explored a few graph-theoretic concepts. Each of these types of ideas has been proposed by suitable examples. We have also investigated the definitions of order, size, cardinality in m-PIVFG. We have explained DN, isolated vertex, total DS, independent set of domination in m-PIVFG with suitable examples. Various important results regarding these dominations have been illustrated. Weak and strong domination, split and non-split domination, strong non-split domination, cycle and path non-split domination, co-total and global non-split domination in m-PIVFG have been explained with supporting examples and important results. An illustrative example from the field of disaster management problem has been described for demonstrating the real-world example of domination on m-PIVFG shows our approach.

We should feature that regarding this investigation, there are distinctive developing regions that we need not demonstrate here as they are outside of our feasible region. In any case, there can be interesting points for future research; for example, one may examine the m-PIVFG with various kinds of environments, e.g., Pythagorean graph, qrung fuzzy graph, fuzzy soft graph, competition graph [12] . Such huge numbers of research bearings stay open; thus, researchers can investigate the problem by the various fuzzy graphs. Later on, we will examine different outcomes of m-PIVFG and expand them to describe various dynamic issues and covering problems under different uncertainties.

Conflict of interest The authors declare that they have no conflict of interest.

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