Global Offensive Alliance in Strong Neutrosophic Graphs

New setting is introduced to study the global offensive alliance. Global offensive alliance is about a set of vertices which are applied into the setting of neutrosophic graphs. Neighborhood has the key role to define this notion. Also, neighborhood is defined based on strong edges. Strong edge gets a framework as neighborhood and after that, too close vertices have key role to define global offensive alliance based on strong edges. The structure of set is studied and general results are obtained. Also, some classes of neutrosophic graphs containing complete, empty, path, cycle, star, and wheel are investigated in the terms of set, minimal set, number, and neutrosophic number. Neutrosophic number is defined in new way. It’s first time to define this type of neutrosophic number in the way that, three values of a vertex are used and they’ve same share to construct this number. It’s called “modified neutrosophic number”. Summation of three values of vertex makes one number and applying it to a set makes neutrosophic number of set. This approach facilitates identifying minimal set and optimal set which forms minimal-global-offensive-alliance number and minimal-global-offensive-alliance-neutrosophic number. Two different types of sets namely global-offensive alliance and minimal-global-offensive alliance are defined. Global-offensive alliance identifies the sets in general vision but minimal-global-offensive alliance takes focus on the sets which deleting a vertex is impossible. Minimal-global-offensive-alliance number is about minimum cardinality amid the cardinalities of all minimal-global-offensive alliances in a given neutrosophic graph. New notions are applied in the settings both individual and family. Family of neutrosophic graphs is studied in the way that, the family only contains same classes of neutrosophic graphs. Three types of family of neutrosophic graphs including m-family of neutrosophic stars with common neutrosophic vertex set, m-family of odd complete graphs with common neutrosophic vertex set, and m-family of odd complete graphs with common neutrosophic vertex set are studied. The results are about minimal-global-offensive alliance, minimal-global-offensive-alliance number and its corresponded sets, minimal-global-offensive-alliance-neutrosophic number and its corresponded sets, and characterizing all minimal-global-offensive alliances. The connection of global-offensive-alliances with dominating set and chromatic number are obtained. The number of connected components has some relations with this new concept and it gets some results. Some classes of neutrosophic graphs behave differently when the parity of vertices are different and in this case, path, cycle, and complete illustrate these behaviors. Two applications concerning complete model as individual and family, under the titles of time table and scheduling conclude the results and they give more clarifications. 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 deeply and with more results but it seems that analogous results are determined. Slight progress is obtained in the family of these models but there are open avenues to study family of other models as same models and different models. 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 strong edges illustrate open way to get results. A set is global offensive alliance when two sets partitioning vertex set have uniform structure. All members of set have more amount of neighbors in the set than out of set. It leads us to the notion of global offensive alliance. Different edges make different neighborhoods but it’s used one style edge titled strong edge. These notions are applied into neutrosophic graphs as individuals and family of them. Independent set as an alliance is a special set which has no neighbor inside and it implies some drawbacks for these notions. Finding special sets which are well-known, is an open way to purse this study. Special set which its members have only one neighbor inside, characterize the connected components where the cardinality of its complement is the number of connected components. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this article.


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(iv) minimal-global-offensive-alliance-neutrosophic number of N T G is S is a minimal-global-offensive alliance.
and it's denoted by Γ s .
vs ∈ E implies s and v have different colors.
It implies S is dominating set.
Thus every vertex v ∈ V \ S, has at least one neighbor in S. The only case is about 175 the relation amid vertices in S in the terms of neighbors. It implies there's S ⊆ S such 176 that |S | is chromatic number.

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Proposition 2.5. Let N T G : (V, E, σ, µ) be a strong neutrosophic graph. Then It implies V is global-offensive alliance. For all set of neutrosophic vertices 196 S, S ⊆ V. Thus for all set of neutrosophic vertices S, Proof. (i). Suppose N T G : (V, E, σ, µ) is a strong neutrosophic graph. Let S = V − {x} 204 where x is arbitrary and x ∈ V. where x is arbitrary and x ∈ V.
is only 522 minimal-global-offensive alliance.
; 565 Figure 7. The set of black circles is minimal-global-offensive alliance.
is only minimal-global-offensive alliances.
is minimal-global-offensive alliance.
is minimal-global-offensive alliance.

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(ii), (iii) and (iv) are obvious.   Time Table and Scheduling   795 In this section, two applications for time table and scheduling are provided where the 796 models are complete models which mean complete connections are formed as individual 797 and family of complete models with common neutrosophic vertex set.

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Designing the programs to achieve some goals is general approach to apply on some 799 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 801 importantance to avoid mixing up.  Step 2. (Issue) Scheduling of program has faced with difficulties to differ amid 806 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 808 every section and to assign to relation amid section, three numbers belong unit 809 interval to state indeterminacy, possibilities and determinacy. There's one 810 restriction in that, the numbers amid two sections are at least the number of the 811 relation amid them. Step 4. (Solution) The neutrosophic graph and its global offensive alliance as model, 815 propose to use specific set. Every subject has connection with every given subject. 816 Thus the connection is applied as possible and the model demonstrates full