Vertex Covered Hypergraphs

Abstract

In this Paper I consider a minimal vertex covering sets and minimal strong vertex covering sets. I define Hypergraph, Sub-hypergraph, Partial sub-hypergraph, Vertex covering set, Minimal Vertex covering set, upper vertex covering set and upper vertex covering number in this paper and I consider the effect of removing a vertex on the upper vertex covering number of the hypergraph. Then I prove that the upper vertex covering number does not increase when a vertex is removed from the hypergraphs and I define upper strong vertex set and upper strong vertex covering number. Here again I consider the effect of removing a vertex from the hypergraph on the strong vertex covering number of the hypergraph. Let G be a hypergraph and S⊆V(G) be a vertex covering set of G. Then S
is a minimal vertex covering set if and only if for every vertex v in S there is an edge f such that S∩f={v}. And I Prove Let G be a hypergraph and v∈V(G). If αb (G\v)<αb (G) then there is a αb-set of G say T such that v∈T. Let G be a hypergraph and v∈V(G). Then αb (G\v)=αb (G) if and only if every αb-set of G \ v is a αb-set of G. Let G be a hypergraph and v∈V(G). Now I prove that if a vertex is removed from the hypergraph and if the upper vertex covering number decrease, then it is exactly 1 decrease. It means Suppose there is an αb-set T of G such that v∈T. If αb (G\v)<αb (G) then αb (G\v)=αb (G)-1.