1- Our work compared with previous research and topics and the difference between them. 2- In this paper we depended the linking between a graph theory and topology on vertices of the graph via the definition of non-neighborhood between the vertices. 3- We introduced new definitions of (semi, pre, semi-pre, b) by depending on the vertices graph. 4- We have studied many theorems of these concepts topological as well as the relationships between them. 5- We have presented in this research many examples to explain. 8- Modern sources were depending on in this research because the topic of linking the graph theory and topology is one of the modern topics
Due it difficult to find applications in topological spaces, which are branches of pure mathematics. The importance of this paper is to find applications in graph theory. So, We Introduced (semi, pre, b, semi-pre)-open subgraph to graph theory. The relations among them were studied. At least many theorems were proofed as a characterization and some examples introduced to explain the subject.
Graph theory, Subgraph, Topology, Closure, Interior.
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