Dominance Matrix of Hypergraph and Its Application in Cyclic Graph Judgement

Abstract

Dominance matrix of soft sets which are finite subsets systems plays an important role in information representation and parameter reduction. This paper bring the concept of dominance matrix to the field of hypergraphs, which can also be taken as a kind of finte subsets systems. We are interested in what it can inspire for the hypergraph theory. Because the $e_{i}\subseteq e_{j}$ condition of acyclic axiom of hypergraphs means that e <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> is dominated by $e_{j}$,this brings a distinct characteristic when to the dominance matrix. We extended the Graham reduction which is used for judging whether a hypergraph is acyclic or not to the domiance matrix of hypergraphs. We prove that a hypergraph is acyclic if and only if the Graham reduction of its dominance matrix comes finally to an empty matrix. We try to show that dominance matrix of hypergraphs can provide a new tool or platform for studying hypergraph theory. Some properties of the cyclic hypergraph are mined, and an application example of underground railway systems is investigated.