Dual-Dominating Numbers in Neutrosophic Setting and Crisp Setting Obtained From Classes of Neutrosophic Graphs

New setting is introduced to study dual-dominating number and neutrosophic dual-dominating number arising from dominated vertices in neutrosophic graphs assigned to neutrosophic graphs. Maximum number of dominated vertices, is a number which is representative based on those vertices. Maximum neutrosophic number of dominated vertices corresponded to dual-dominating set is called neutrosophic dual-dominating number. Forming sets from dominated vertices to figure out different types of number of vertices in the sets from dominated sets n in the terms of maximum number of vertices to get maximum number to assign to neutrosophic graphs is key type of approach to have these notions namely dual-dominating number and neutrosophic dual-dominating number arising from dominated vertices in neutrosophic graphs assigned to neutrosophic graphs. Two numbers and one set are assigned to a neutrosophic graph, are obtained but now both settings lead to approach is on demand which is to compute and to find representatives of sets having largest number of dominated vertices from different types of sets in the terms of maximum number and maximum neutrosophic number forming it to get maximum number to assign to a neutrosophic graph. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then for given two vertices, s and n, if μ(ns) = σ(n) ∧ σ(s), then s dominates n and n dominates s. Let S be a set of neutrosophic vertices [a vertex alongside triple pair of its values is called neutrosophic vertex.]. If for every neutrosophic vertex s in S, there’s at least one neutrosophic vertex n in V \ S such that n dominates s, then the set of neutrosophic vertices, S is called dual-dominating set. The maximum cardinality between all dual-dominating sets is called dual-dominating number and it’s denoted by D(NTG); for given two vertices, s and n, if μ(ns) = σ(n) ∧ σ(s), then s dominates n and n dominates s. Let S be a set of neutrosophic vertices [a vertex alongside triple pair of its values is called neutrosophic vertex.]. If for every neutrosophic vertex s in S, there’s at least one neutrosophic vertex n in V \ S such that n dominates s, then the set of neutrosophic vertices, S is called dual-dominating set. The maximum neutrosophic cardinality between all dual-dominating sets is called neutrosophic dual-dominating number and it’s denoted by Dn(NTG). As concluding results, there are some statements, remarks, examples and clarifications about some classes of neutrosophic graphs namely path-neutrosophic graphs, cycle-neutrosophic graphs, complete-neutrosophic graphs, star-neutrosophic graphs, complete-bipartite-neutrosophic graphs, complete-t-partite-neutrosophic graphs, and wheel-neutrosophic graphs. The

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clarifications are also presented in both sections “Setting of dual-dominating number,” and “Setting of neutrosophic dual-dominating number,” for introduced results and used classes. This approach facilitates identifying sets which form dual-dominating number and neutrosophic dual-dominating number arising from dominated vertices in neutrosophic graphs assigned to neutrosophic graphs. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. The cardinality of set of dominated vertices and neutrosophic cardinality of set of dominated vertices corresponded to dual-dominating set have eligibility to define dual-dominating number and neutrosophic dual-dominating number but different types of set of dominated vertices to define dual-dominating sets. Some results get more frameworks and perspective about these definitions. The way in that, different types of set of dominated vertices in the terms of maximum number to assign to neutrosophic graphs, opens the way to do some approaches. 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. Neutrosophic dual-dominating notion is applied to different settings and classes of neutrosophic graphs. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this article.