Three Types of Neutrosophic Alliances based on Connectedness and (Strong) Edges

New setting is introduced to study the alliances. Alliances are about a set of vertices which are applied into the setting of neutrosophic graphs. Neighborhood has the key role to define these notions. Also, neighborhood is defined based on the edges, strong edges and some edges which are coming from connectedness. These three types of edges get a framework as neighborhood and after that, too close vertices have key role to define offensive alliance, defensive alliance, t-offensive alliance, and t-defensive alliance based on three types of edges, common edges, strong edges and some edges which are coming from connectedness. The structure of set is studied and general results are obtained. Also, some classes of neutrosophic graphs containing complete, empty, path, cycle, bipartite, t-partite, star and wheel are investigated in the terms of set, minimal set, number, and neutrosophic number. In this study, there’s an open way to extend these results into the family of these classes of neutrosophic graphs. The family of neutrosophic graphs aren’t study but it seems that analogous results are determined. There’s a question. How can be related to each other, two sets partitioning the vertex set of a graph? The ideas of neighborhood and neighbors based on different edges illustrate open way to get results. A set is alliance when two sets partitioning vertex set have uniform structure. All members of set have different amount of neighbors in the set and out of set. It leads us to the notion of offensive and defensive. New ideas, offensive alliance, defensive alliance, t-offensive alliance, t-defensive alliance, strong offensive alliance, strong defensive alliance, strong t-offensive alliance, strong t-defensive alliance, connected offensive alliance, connected defensive alliance, connected t-offensive alliance, and connected t-defensive alliance are introduced. Two numbers concerning cardinality and neutrosophic cardinality of alliances are introduced. A set is alliance when its complement make a relation in the terms of study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this article.

[4], are proposed. 6 In this section, I use two subsections to illustrate a perspective about the 7 background of this study. In this study, there's an idea which could be considered as a motivation. 10 Question 1.1. Is it possible to use mixed versions of ideas concerning "alliance", 11 "offensive" and "defensive" to define some notions which are applied to neutrosophic 12 graphs? 13 It's motivation to find notions to use in any classes of neutrosophic graphs.
14 Real-world applications about time table and scheduling are another thoughts which 15 lead to be considered as motivation. Connections amid two vertices have key roles to 16 assign alliances, defensive alliances and offensive alliances. Thus they're used to define 17 new ideas which conclude to the structure alliances, defensive alliances and offensive 18 alliances. The concept of having general edge inspires me to study the behavior of 19 general, strong edges and connected edge in the way that, three types of numbers and 20 set, e.g., alliances, defensive alliances and offensive alliances are the cases of study in the 21 settings of individuals and in settings of families. Also, there are some extensions into 22 alliances, t-defensive alliances and t-offensive alliances. 23 The framework of this study is as follows. In the beginning, I introduced basic 24 definitions to clarify about preliminaries. In subsection "Preliminaries", new notions of 25 (strong/connected)alliances, (strong/connected)t-defensive alliances and 26 (strong/connected)t-offensive alliances are applied to set of vertices of neutrosophic 27 graphs as individuals. In section "In the Setting of Set", specific sets have the key role 28 in this way. Classes of neutrosophic graphs are studied in the terms of different sets in 29 section "Classes of Neutrosophic Graphs" as individuals. In the section "In the Setting 30 of Number", usages of general numbers have key role in this study as individuals. In

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Based on different edges, it's possible to define different neighbors as follows.
Definition 1.9. Let N T G : (V, E, σ, µ) be a neutrosophic graph. Then . V is strong offensive alliance since the following statements are equivalent.
. V is connected offensive alliance since the following statements are equivalent.
. V is offensive alliance since the following statements are equivalent.

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(v). V is strong offensive alliance since the following statements are equivalent.  (vi). V is connected offensive alliance since the following statements are equivalent. Proof. Suppose N T G : (V, E, σ, µ) is a neutrosophic graph. Consider ∅. All members of 150 ∅ have no neighbor inside the set less than neighbor out of set. Thus, 151 (i). ∅ is defensive alliance since the following statements are equivalent.
. ∅ is strong defensive alliance since the following statements are equivalent.
. ∅ is connected defensive alliance since the following statements are equivalent.

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(v). ∅ is strong defensive alliance since the following statements are equivalent.

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(vi). ∅ is connected defensive alliance since the following statements are equivalent.
. An independent set is strong defensive alliance since the following statements are 210 equivalent.
. An independent set is connected defensive alliance since the following statements 218 are equivalent.

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(iv). An independent set is defensive alliance since the following statements are 226 equivalent.

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(v). An independent set is strong defensive alliance since the following statements are 233 equivalent.

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(vi). An independent set is connected defensive alliance since the following statements 240 are equivalent.
Thus it's contradiction. It implies every V \ {x} isn't offensive alliance in a given cycle. 267 Consider one vertex is out of S which is alliance. This vertex has one neighbor in S, i.e, 268 Suppose x ∈ V \ S such that y, z ∈ N (x). By it's path, |N (x)| = |N (y)| = |N (z)| = 2. 269 Thus Thus it's contradiction. It implies every V \ {x} isn't offensive alliance in a given path. 277 Consider one vertex is out of S which is alliance. This vertex has one neighbor in Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 January 2022 ∃y ∈ S, 1 < 1.

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Thus it's contradiction. It implies every V \ {x} isn't offensive alliance in a given wheel. 287 (ii), (iii) are obvious by (i).

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(iv). By (i), |V | is minimal and it's offensive alliance. Thus it's |V |-offensive alliance. 289 (v), (vi) are obvious by (iv).  Proof. Suppose N T G : (V, E, σ, µ) is a neutrosophic graph which is cycle/path//wheel. 299 (i). Consider one vertex is out of S which is alliance. This vertex has one neighbor 300 in S, i.e, Suppose x ∈ V \ S such that y, z ∈ N (x). By it's cycle,   V is offensive alliance since the following statements are equivalent.

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V is strong offensive alliance since the following statements are equivalent.

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V is connected offensive alliance since the following statements are equivalent.

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V is offensive alliance since the following statements are equivalent.

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V is strong offensive alliance since the following statements are equivalent.

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V is connected offensive alliance since the following statements are equivalent.  The number is 0 and the neutrosophic number is 0, for an independent set in the setting 546 of offensive alliance.

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(ii). ∅ is strong defensive alliance since the following statements are equivalent.
The number is 0 and the neutrosophic number is 0, for an independent set in the setting 555 of strong offensive alliance.

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(iii). ∅ is connected defensive alliance since the following statements are equivalent.
The number is 0 and the neutrosophic number is 0, for an independent set in the setting 564 of connected offensive alliance.

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(iv). ∅ is defensive alliance since the following statements are equivalent. The number is 0 and the neutrosophic number is 0, for an independent set in the setting 572 of 0-offensive alliance.

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(v). ∅ is strong defensive alliance since the following statements are equivalent. The number is 0 and the neutrosophic number is 0, for an independent set in the setting 588 of connected 0-offensive alliance.

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Then there's no independent set.  Let G be a family of N T Gs : (V, E, σ, µ) neutrosophic graphs which 675 are from one-type class which the result is obtained for individual. Then results also hold 676 for family G of these specific classes of neutrosophic graphs.

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Proof. There are neither conditions nor restrictions on the vertices. Thus the result on 678 individual is extended to the result on family. 679 6 Applications in Time Table and Scheduling   680 Designing the programs to achieve some goals is general approach to apply on some 681 issues to function properly. Separation has key role in the context of this style.

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Separating the duration of work which are consecutive, is the matter and it has 683 important to avoid mixing up.  Step 2. (Issue) Scheduling of program has faced with difficulties to differ amid 688 consecutive section. Beyond that, sometimes sections are not the same.

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Step 3. (Model) The situation is designed as a model. The model uses data to assign 690 every section and to assign to relation amid section, three numbers belong unit 691 interval to state indeterminacy, possibilities and determinacy. There's one 692 restriction in that, the numbers amid two sections are at least the number of the 693 relation amid them.