0049 | (Failed)1-Zero-Forcing Number in Neutrosophic Graphs

New setting is introduced to study 1-zero-forcing number, 1-zero-forcing neutrosophic-number, failed 1-zero-forcing number and failed 1-zero-forcing neutrosophic-number arising operations of different vertices. Leaf-like is a key term to have these notions. (Not) Forcing a vertex to change its color is a type of approach to force that vertex to be zero-like. (Not) Forcing a vertex which is only neighbor for zero-like vertex to be zero-like vertex but now both settings leads to approach is on demand which is finding biggest (smallest) set which doesn’t force. Let

NTG : (V,E,σ,μ) be a neutrosophic graph. Then 1-zero-forcing number Z(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is minimum cardinality of a set S of black vertices (whereas vertices in V (G) \ S are colored white) such that V (G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. The last condition is as follows. For one time, black can change any vertex from white to black. 1-zero-forcing neutrosophic-number Zn(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is minimum neutrosophic cardinality of a set S of black vertices (whereas vertices in

V (G) \ S are colored white) such that V (G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. The last condition is as follows. For one time, black can change any vertex from white to black. Failed 1-zero-forcing number Z′(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximum cardinality of a set S of black vertices (whereas vertices in V (G) \ S are colored white) such that V (G) isn’t turned black after finitely many applications of “the color-change rule”: a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. The last condition is as follows. For one time, Black can change any vertex from white to black. The last condition is as follows. For one time, black can change any vertex from white to black. Failed 1-zero-forcing neutrosophic-number Zn′ (NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximum neutrosophic cardinality of a set S of black vertices (whereas vertices in V (G) \ S are colored white) such that V (G) isn’t turned black after finitely many applications of “the color-change rule”: a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. The last condition is as follows. For one time, Black can change any vertex from white to black. The last condition is as follows. For one time, black can change any vertex from white to black. 1-zero-forcing number, 1-zero-forcing neutrosophic-number, failed 1-zero-forcing number and failed 1-zero-forcing neutrosophic-number arising operations of different vertices are about a set of vertices which are applied into the setting of

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neutrosophic graphs. The structure of set is studied and general results are obtained. Also, some classes of neutrosophic graphs namely path-neutrosophic graphs, cycle-neutrosophic graphs, complete-neutrosophic graphs, star-neutrosophic graphs, bipartite-neutrosophic graphs, and t-partite-neutrosophic graphs are investigated in the terms of maximal set minimal set which form 1-zero-forcing number, 1-zero-forcing neutrosophic-number, failed 1-zero-forcing number and failed 1-zero-forcing neutrosophic-number arising operations of different vertices. Neutrosophic number is reused in this way. It’s applied to use the type of neutrosophic number in the way that, three values of a vertex are used and they’ve same share to construct this number to compare with other vertices. Summation of three values of vertex makes one number and applying it to a comparison. This approach facilitates identifying vertices which form 1-zero-forcing number, 1-zero-forcing neutrosophic-number, failed 1-zero-forcing number and failed 1-zero-forcing neutrosophic-number arising operations of different vertices. In path-neutrosophic graphs, ∅ (1-set), forms maximal (minimal) set but with slightly differences, in cycle-neutrosophic graphs, ∅ (1-set), forms maximal (minimal) set. Other classes have same approaches. In complete-neutrosophic graphs, a set of vertices excluding two (three) vertices leads us to (failed) 1-zero-forcing number and (failed) 1-zero-forcing neutrosophic-number. In star-neutrosophic graphs, a set of vertices excluding only two (three) vertices and containing center, makes (maximal) minimal set. In complete-bipartite-neutrosophic graphs, a set of vertices excluding three (four) vertices from (same) different parts as possible makes intended set but with slightly differences, in complete-t-partite-neutrosophic graphs, a set of vertices excluding three (four) vertices from (same) different parts makes intended set. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. Using basic set (not) to extend this set to set of all vertices has key role to have these notions in the form of 1-zero-forcing number, 1-zero-forcing neutrosophic-number, failed 1-zero-forcing number and failed 1-zero-forcing neutrosophic-number arising operations of different vertices. The cardinality of a set has eligibility to form (failed) 1-zero-forcing number but the neutrosophic cardinality of a set has eligibility to call (failed) 1-zero-forcing neutrosophic-number. Some results get more frameworks and perspective about these definitions. The way in that, three vertices (don’t) have unique connection amid each other, opens the way to do some approaches. A vertex could affect on other vertex but there’s no usage of edges. These notions are applied into neutrosophic graphs as individuals but not family of them as drawbacks for these notions. Finding special neutrosophic graphs which are well-known, is an open way to pursue this study. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this article.