Mohammad Hamidi - Department of Mathematics, University of Payame Noor, P.O.Box ۱۹۳۹۵-۴۶۹۷, Tehran, Iran.

A hypertree is a special type of connected hypergraph that removes‎ ‎any‎, ‎its hyperedge then results in a disconnected hypergraph‎. ‎Relation between hypertrees (hypergraphs) and trees (graphs) can be helpful to solve real problems in hypernetworks and networks and it is the main tool in this regard‎. ‎The purpose of this paper is to introduce a positive relation (as \alpha-relation) on hypertrees that makes a connection between hypertrees and trees‎. ‎This relation is dependent on some parameters such as path‎, ‎length of a path‎, ‎and the intersection of hyperedges‎. ‎For any q\in \mathbb{N}‎, ‎ we introduce the concepts of a derivable tree‎, ‎(\alpha‎, ‎q)-hypergraph‎, ‎and fundamental (\alpha‎, ‎q)-hypertree for the first time in this study and analyze the structures of derivable trees from hypertrees via given positive relation‎. ‎In the final‎, ‎we apply the notions of derivable trees‎, ‎(\alpha‎, ‎q)-trees in real optimization problems by modeling hypernetworks and networks based on hypertrees and trees‎, ‎respectively.‎‎‎

‎\alpha-Relation, Fundamental (\alpha, q)-hypergraph, k-Parts