Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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

Version 1 : Received: 15 January 2022 / Approved: 17 January 2022 / Online: 17 January 2022 (15:29:26 CET)

How to cite: Garrett, H. Three Types of Neutrosophic Alliances based on Connectedness and (Strong) Edges. Preprints 2022, 2022010239 (doi: 10.20944/preprints202201.0239.v1). Garrett, H. Three Types of Neutrosophic Alliances based on Connectedness and (Strong) Edges. Preprints 2022, 2022010239 (doi: 10.20944/preprints202201.0239.v1).

Abstract

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 neighborhood. Different edges make different neighborhoods. Three types of edges are applied to define three styles of neighborhoods. General edges, strong edges and connected edges are used where connected edges are the edges arising from connectedness amid two endpoints of the edges. 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 this 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.

Keywords

Alliance; Offensive Alliance; Defensive Alliance

Subject

MATHEMATICS & COMPUTER SCIENCE, Applied Mathematics

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