Strong Subgraph K-connectivity Of Two Types Of Directed Graph

The generalized k-connetivity κk(G)of a graph G which was introduced by Hager in 1985,is a natural generalization of the path-version definition of the classic vertex connectivity.Sun,Gutin,Yeo,and Zhang replaced”S-tree" in the definition of generalized connectivity with"connected subgraphs containing S in G",and then replaced the condition of"connected" with"strongly connected" to extended the generalized k-connectivity of graphs,they define the strong subgraph kconnectivity as follow.Let D=(V,A)be a digraph of order n,S is a k-subset of V(D)(2≤k≤n).let D1,D2,...,Dp which containing S are strong subgraphs of D.For any subgraph Di,Dj(i,j ∈[p]),we said they are internal disjoint S-strong subgraphs if V(Di)∩ V(Dj)=S and A(Di)∩ A(Dj)=?.κS(D)is defined as the maximum number of internal disjoint S-strong subgraphs,and strong subgraph k-connectivity is defined asκk(D)=min{κS(D)|S?V(D),|S|=k}.In this article,we mainly study the strong subgraph k-connectivity of Cartesian product digraphs and complete bipartite digraph.In chapter 1,we introduce the research background and current status of the strong subgraph connectivity of digraph.In chapter 2,the exact values of the strong subgraph 2-connectivity and strong subgraph 3connectivity of some Cartesian product digraphs are given.In chapter 3,we find the bounds on the strong subgraph k-connectivity of any complete bipartite digraph(?)a,b and determine the exact value of κk((?)a,b)for k=2 and k=a+b.The chapter 4,we summarizes the research results of this article and gives the contents for further research.