Extract mesh vertices + Create a Quad Mesh whose vertices have a connectivity with 8 neighbours

I am trying to create a mesh from a surface to use its vertices for a relaxation. I created a mesh from lines using weaverbird. However, weaverbird and deconstruct-mesh component did not give me all vertices and faces. Could you have any solution for this problem? Are there better ways to do it?

MeshFrLines_re.gh (10.6 KB)

I think you could do it without weaverbird, with mesh from points:

thanks Kim!

thanks Baris

I understood that I need a diagonal mesh. Because I want to find 8 neighbours of each vertex.
Do you have any suggestion how to create a mesh whose nodes have 8 neighbours?
@HS_Kim @Baris

MeshFrLines_reV2.gh (8.8 KB)

thanks Kim:)

Kim @HS_Kim , now the problem is the code creates vertex at the middle of each face. can I get rid off the vertices in the middle of the faces? or is there any other way to define a quad mesh but each vertices has a connection with 8 neighbours?