Abstract
Graphs serve as one of the main tools for the mathematical modeling of various human problems. Fuzzy graphs have the ability to solve uncertain and ambiguous problems. The cubic graph, which has recently gained a position in the fuzzy graph family, has shown good capabilities when faced with problems that cannot be expressed by fuzzy graphs and interval-valued fuzzy graphs. Simultaneous application of fuzzy and interval-valued fuzzy membership indicates a high flexibility in modeling uncertainty issues. The vertex cover is a fundamental issue in graph theory that has wide application in the real world. The previous definition limitations in the vertex covering of fuzzy graphs has directed us to offer new classifications in terms of cubic graph. In this study, we introduced the strong vertex covering and independent vertex covering in a cubic graph with strong edges and described some of its properties. One of the motives of this research was to examine the changes in the strong vertex covering number of a cubic graph if one vertex is omitted. This issue can play a decisive role in covering the graph vertices. Since many of the problems ahead are of hybrid type, by reviewing some operations on the cubic graph we were able to determine the strong vertex covering number on the most important cubic product operations. Finally, two applications of strong vertex covering and strong vertex independence are presented.
Highlights
Graphs have long been used to describe objects and the relationships among them
The vertex cover (VC) of a graph G is an arrangement of vertices, in which every edge in G has at least one end point in this set provided that each vertex in G is at least adjacent to one edge
Fuzzy theory is one of the best and most powerful tools for modeling problems in examining the relationships among uncertainties in the real world. This concept gained popularity with the introduction of the fuzzy set by Zadeh [4], and fuzzy graph (FG) by Rosenfeld [5], as they are characterized by two membership functions in [0, 1] for vertices and edges of a graph
Summary
Graphs have long been used to describe objects and the relationships among them. The vertex cover (VC) of a graph G is an arrangement of vertices, in which every edge in G has at least one end point in this set provided that each vertex in G is at least adjacent to one edge. Fuzzy theory is one of the best and most powerful tools for modeling problems in examining the relationships among uncertainties in the real world This concept gained popularity with the introduction of the fuzzy set by Zadeh [4], and fuzzy graph (FG) by Rosenfeld [5], as they are characterized by two membership functions in [0, 1] for vertices and edges of a graph. Jun et al [33] introduced the idea of the cubic set (CS) in the form of a combination of FS and IVFS, serving as a more general tool for modeling uncertainty and ambiguity Through applying this concept, we can solve various problems instigated by uncertainties and have the best choice using CSs in decision-making. The CG in graph theory is referred to as 3-regular graphs, the meaning of the CG throughout this article means a cubic fuzzy graph consisting of an IVFG and an FG
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