Basic Maximal (m+1)K₂-free Bipartite Graph

A matching M of G is induced if E(V(M))=M.The induced matching number of G,denoted by IM(G),is the number of a maximum induced matching of G.H is called a proper induced subgraph of G if H is an induced subgraph of G with H≠G.We call G a maximal(m+1)K₂-free bipartite Graph,if G is a connected but not complete simple bipartite graph,such that for each pair of nonadjacent vertices x and y with G+xy being bipartite graph,we have IM(G+xy)=IM(G)+1=m+1.We say G is a basic maximal(m+1)K₂-free bipartite Graph,if G is a maximal (m+1)K₂-free bipartite Graph without any proper induced subgraph of G being a maximal(n+1)K₂-free bipartite Graph for any n∈N.We have found some maximal 2K₂-free graphs of order 12,13,14,15 and 16.In this paper,we will consider the structure of a basic maximal(m+1)K₂-free bipartite graph.For a basic maximal(m+1)K₂-free bipartite graph G,We characterize the number of vertices,the maximum degree,the connectivity and the minimum degree.We consider a basic maximal(m+1)K₂-free bipartite graph G with a vertex cut T={u₁,u₂}specially and characterize the number of vertices,the maximum degree and the minimun degree of a component of G-T.