Nikfar Domination Versus Others: Restriction, Extension Theorems and Monstrous Examples

The aim of this expository article is to present recent developments in the centuries-old discussion on the interrelations between several types of domination in graphs. However, the novelty even more prominent in the newly discovered simplified presentations of several older results. Domination can be seen as arising from real-world application and extracting classical results as first described by this article.The main part of this article, concerning a new domination and older one, is presented in a narrative that answers two classical questions: (i) To what extend must closing set be dominating? (ii) How strong is the assumption of domination of a closing set? In a addition, we give an overview of the results concerning domination. The problem asks how small can a subset of vertices be and contain no edges or, more generally how can small a subset of vertices be and contain other ones. Our work was as elegant as it was unexpected being a departure from the tried and true methods of this theory that had dominated the field for one fifth a century. This expository article covers all previous definitions. The inability of previous definitions in solving even one case of real-world problems due to the lack of simultaneous attentions to the worthy both of vertices and edges causing us to make the new one. The concept of domination in a variety of graphs models such as crisp, weighted and fuzzy, has been in a spotlight. We turn our attention to sets of vertices in a fuzzy graph that are so close to all vertices, in a variety of ways, and study minimum such sets and their cardinality. A natural way to introduce and motivate our subject is to view it as a real-world problem. In its most elementary form, we consider the problem of reducing waste of time in transport planning. Our goal here is to first describe the previous definitions and the results, and then to provide an overview of the flows ideas in their articles. The final outcome of this article is twofold: (i) Solving the problem of reducing waste of time in transport planning at static state; (ii) Solving and having a gentle discussions on problem of reducing waste of time in transport planning at dynamic state. Finally, we discuss the results concerning holding domination that are independent of fuzzy graphs. We close with a list of currently open problems related to this subject. Most of our exposition assumes only familiarity with basic linear algebra, polynomials, fuzzy graph theory and graph theory.

1 Introduction and Overview 1 Domination are among the most fundamental concepts of graph theory. Also, 2 domination can behave in many strange ways. For instance, besides the classical 3 definitions of domination, there are many characterization of this concept. One of this 4 characterization due to A. Somasundaram and S. Somasundaram (Ref. [31]), see also 5 Refs. [8, 13-15, 17, 22-24, 30, 32] for further generalizations. One the contrary and quite 6 surprisingly, there are nowhere these definitions Solving the problem of reducing waste 7 of time in transport planning and also (separately) all others real-world problems see 6. 8 Somehow, a key direction of study of domination deals with trying to provide a clear 9 structure of what the dominating set of vertices looks like. The leading theme of this 10 expository article is to discuss the following two questions concerning fuzzy graphs 11 Q1: How much closing does dominating imply? 12 Q2: How much dominating does closing imply? 13 They will be addressed in sections 2 and 6, respectively. The main narrative 14 presented in these sections is independent of any results from graph theory and/or 15 calculus. The purpose of this expository article is to provide an overview of the authors' 16 recent series of work (Refs. [8, 13-15, 17, 22-24, 30-32]), in which a positive answer to the 17 problem of reducing waste of time in transport planning for the our new definition is 18 given. 19 Consider a set of cities connected by communication paths, Which cities is connected 20 to others by roads? We face with a graph model of this situation. But the cities are not 21 same and they have different privileges in low traffic levels and this events also occur for 22 the roads in low-cost levels. So we face with the weighted graph model, at first. These 23 privileges are not crisp but they are vague in nature. So we don't have a weighted graph 24 model. In other words, we face with a fuzzy graph model, which must study the concept 25 of domination on it. 26 Next we turn our attention to sets of vertices in a fuzzy graph G that are close to all 27 vertices of G, in a variety of ways, and study minimum such sets and their cardinality. 28 In 1998, the concept of effective domination in fuzzy graphs was introduced by A. 29 Somasundaram and S. Somasundaram (Ref. [31]) as the classical problems of covering 30 chess board with minimum number of chess pieces. In 2010, the concept of 31 2-strong(weak) domination in fuzzy graphs was introduced by C. Natarajan and S.K. 32 with strong edges. In 2015, the concept of 2-domination in fuzzy graphs was introduced 36 by A. Nagoor Gani and K. Prasanna Devi (Ref. [22]) as the extension of 2-domination 37 in crisp graphs. In 2015, the concept of strong domination in fuzzy graphs was 38 introduced by O.T. Manjusha and M.S. Sunitha (Ref. [13]) as reduction of the value of 39 old domination number and extraction of classic results. In 2016, the concept of 40 (1, 2)−domination in fuzzy graphs was introduced by N. Sarala and T. Kavitha 41 (Ref. [30]) as the extension of (1, 2)−domination in crisp graphs. A few researchers 42 studied other domination variations which are based on above definitions. So we only 43 compare our new definition with the fundamental dominations.

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A fuzzy set on a given set V, is a map assigning to every its elements a real number 45 from unit interval [0, 1]; this number is called value of element in V.

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The number of a fuzzy set is a summation on values of all its elements.

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A fuzzy graph G is an ordered pair (V, E) consisting of a fuzzy set V of vertices and 48 a fuzzy set E, disjoint from V, of edges, together with an incidence function φ G that 49 associates with each edge of G an unordered pair of (not necessarily distinct) vertices of 50 G. If e is an edge and u and v are vertices such that φ G (e) = {u, v}, then e is said to 51 join u and v, and the vertices u and v are called the ends of e. In a fuzzy graph, value 52 of every vertices are at least equal to value of their ends. We denote the numbers of 53 vertices and edges in G by n(V ) and n(E); these two basic parameters are called the 54 order and size of G, respectively. In section ??, some examples should serve to clarify 55 the definition. For notational simplicity, we write uv for the unordered pair {u, v}. 56 We don't speak about a graph. So when we write vertices or edges, we talk about a 57 fuzzy graph.

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A path is a sequence of vertices v 0 v 1 · · · v n such that v i−1 v i is an edge for any 59 1 ≤ i ≤ n. The least value between edges in a path is called its value. In other words, if 60 e has the least value in a path P , we would call E(e) value of path and it is denoted by 61 the same notation E(P ). In this case, we have E(e) = E(P ). The greatest value 62 between all paths from the vertices x to y in a fuzzy graph G = (V, E) is called value 63 between x and y and is denoted by E(x, y).

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A fuzzy graph G = (V, E) is connected if for every x, y in V, E G (x, y) > 0. 65 Note that E G−xy (x, y) is the strength of connectedness between x and y in the fuzzy 66 graph obtained from G by deleting the edge xy. An edge xy in G is α-strong if 67 E(xy) > E G−xy (x, y). An edge xy in G is β-strong if E(xy) = E G−xy (x, y). An edge xy 68 is a strong edge if it is either α−strong or β−strong. An edge uv of a fuzzy graph is 69 called an M -strong edge, In order to avoid confusion with the notion of strong edges, we 70 shall call strong in the sense of Mordeson as M-strong, if E(uv) = V (u) ∧ V (v). If 71 E(uv) > 0, then u and v are called the neighbors. The set of all neighbors of u is 72 denoted by N (u). Also v is called the α-strong neighbor of u, if the edge uv is α-strong. 73 The set of all α-strong neighbors of u is denoted by N s (u). The degree of a vertex v is   The complement of a fuzzy graph G = (V, E) denoted byḠ, is defined to bipartite fuzzy graph and is denoted by K V1,V2 , where V 1 and V 2 are respectively the 88 restrictions of V to V 1 and V 2 . In this case, If either |V 1 | = 1 or |V 2 | = 1 then the 89 complete bipartite fuzzy graph is said a star fuzzy graph which is denoted by Now, we will define some special operations on fuzzy graphs. The pages of references 92 will show the proof of validity of them.

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The union G = G 1 ∪ G 2 in Ref. ( [19], Proposition 3.1, pp.166,167) of two fuzzy be the set of all 1-strong dominating sets in G. The 1-strong domination number of G is 125 defined by γ Sn (G) = min D∈S (Σ u∈D V (u)).

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(iv) (Ref. [22],   In what follows, we will work under this generality.    Proof. Let G be a complete fuzzy graph. The strength of path P from u to v is of the 221   It is interesting to note the converse of Proposition 3.2, that does not hold. Now, the fuzzy set E is defined by E(v 1 v 2 ) = 0.005, So the set {v 2 , v 3 } is the α-strong dominating set. This set is also nikfar dominating set 236 in fuzzy graph G. Hence γ v (G) = 1.75 + 0.9 + 0.7 = 3.35 = Σ u∈v V (u) = p. Therefore   E(x, y). In other words, if xy is in G, but not F, there is a path in F 269 between x and y whose strength is greater than E(xy). It is clear that a forest is a fuzzy 270 forest. If G is connected, then so is F since any edge of a path in G is either in F, or 271 can be diverted through F. In this case, we call G a fuzzy tree.  Proof. Let G be a fuzzy graph. SoḠ is also fuzzy graph. We implement Theorem 4.1, 295 on G andḠ. Then γ v ≤ p andγ v ≤ p. Hence γ v +γ v ≤ 2p.  Proof. By attentions to all edges between two sets, which are only α-strong, the result 301 follows.

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A domatic partition is a partition of the vertices of a graph into disjoint dominating 303 sets. The maximum number of disjoint dominating sets in a domatic partition of a 304 graph is called its domatic number.

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Finding a domatic partition of size 1 is trivial and finding a domatic partition of size 306 2 (or establishing that none exists) is easy but finding a maximum-size domatic Proof. For every connected fuzzy graph, V is an α-strong dominating set. By analogous 313 to proof of Theorem 4.4, we can obtain the result. 314 We improve the upper bound for the nikfar domination number of fuzzy graphs 315 without isolated vertices, (Theorem 4.6).

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Theorem 4.6. For any fuzzy graph G = (V, E) without isolated vertices, we have Proof. Let D be a minimal dominating set of G. By Theorem 4.5, V-D is an α-strong Hence the proof 321 is completed. 322 We also improve Nordhaus-Gaddum (NG)'s result for fuzzy graphs without isolated 323 vertices, (Corollary 4.7).

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Proof. By the Implement of Theorem 4.6, on G andḠ, we have is a contradiction. Hence the only possible case is γ v =γ v = p 2 .

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Proposition 4.8. Let G = (V, E) be a fuzzy graph. If all edges have equal value, then 333 G has no α-strong edge.

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Proof. By using Definition of α-strong edge, the result is hold.

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The following example illustrates this concept. We give a necessary and sufficient condition for nikfar domination number which is 344 half of order under the conditions. In fact, the fuzzy graphs which whose nikfar 345 domination number is half of order, are characterized under the conditions, (Theorem Hence the 354 result is hold in this case.
The result is hold in this case. The goal of this section is to prove some results concerning operations on a fuzzy graph 362 and study some conjectures arising from it.

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The nikfar domination of union of two fuzzy graphs is studied, (Proposition 5.1).

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Proposition 5.1. Let G 1 and G 2 be fuzzy graphs. The nikfar dominating set of 365 G 1 ∪ G 2 is D = D 1 ∪ D 2 such that D 1 and D 2 are the nikfar dominating sets of G 1 and 366 G 2 , respectively. Moreover, Proof. By using Definition of union of two fuzzy graphs, the result is obviously hold.

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Also the nikfar domination of union of fuzzy graphs family is discussed, (Corollary 369 5.2).

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By using α-strong edge and monotone decreasing fuzzy graph property, the result in 392 relation with Vizing's conjecture is determined, (Theorem 5.8).

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Hence Vizing's conjecture is also hold for G − e. Then the result follows. Vizing's conjecture is also hold for K. So the result follows.

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The nikfar domination of join of two fuzzy graphs is studied, (Proposition 5.10).

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Proof. By using Definition of join of two fuzzy graphs in this case, M -strong edges 411 between two fuzzy graphs is not α-strong which is a weak edge changing strength of 412 connectedness of G.

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Also the nikfar domination of join of fuzzy graphs family is discussed, (Corollary 414 5.11).

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Corollary 5.11. Let G 1 , G 2 , · · · , G n be fuzzy graphs. The nikfar dominating set of It is obvious from the above table and Figure 5 that the desirable cities given by 468 previous definitions, are meaningless due to the lack of simultaneous attention to cities 469 and roads.  Dynamic analysis of networks in the first row of Figure 6 are the following table.