Eulerian and Hamiltonian Soft Semigraphs

Abstract

Soft set theory is a mathematical approach to address the challenges of handling vague or uncertain information. It is a more advanced version of classical set theory that deals with imprecise elements and enables the flexible representation of uncertain data. It involves categorizing the elements of the universe based on specific parameters. Semigraph is a generalization of a graph which is different from a hypergraph. A hypergraph extends the concept of a graph by allowing any subset of vertices to form an edge. Semigraphs, on the other hand, distinguish themselves from hypergraphs by imposing a specific order on the vertices within each edge. Soft semigraphs were developed using the principles of soft set theory applied to semigraphs. This study introduces Eulerian and Hamiltonian soft semigraphs. We establish a necessary and sufficient condition for a soft semigraph to be Eulerian, relying on parameters such as [Formula: see text]-part consecutive adjacent degree, [Formula: see text]-part end degree, and the [Formula: see text]-part consecutive adjacency graph. Additionally, we provide the conditions for a soft semigraph to be Hamiltonian. We introduce the concept of maximal non-Hamiltonian [Formula: see text]-part. Finally, we define the closure of a soft semigraph and demonstrate the relationship between a Hamiltonian soft semigraph and its closure.