1)mesh connectivity using directed acyclic graph 2)mesh approximate surface using mesh connectivity

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Hi there! I'm new here and I'm not from the field of math/computer science. However I always get involved in topics related to it. Please guys, if what I ask is trivial be patient with me

I'm building up an algorithm to construct a mesh approximating a given surface. I guess there are many algorithms out there for this. However, in my case I need this mesh to have a special connectivity. This connectivity emerges when I define a sequence of nodes in which every node may be connected by edges to the next nodes of the sequence but never to the previous ones. I have done a bit of research and in order to construct this connectivity I thought of using a directed acyclic graph since this allows me to construct the sequence of nodes I need. I later wish to construct a mesh approximating a given surface using the defined connectivity. Questions:

1. Can a mesh be still called "mesh" if it has no faces, only edges and vertices?
2. Does it make sense to use the directed acyclic graph for definning the connectivity of a geometric mesh? Is it common to do this? Any reference?
3. Are there well-known methods to construct directed acyclic graphs?
4. Can I construct the mesh approximating a given surface using a defined connectivity? Any well-known method for this? How in simple terms this process works? mapping? Can I do it if the graph is directed or should I first turn the graph into an undirected one? I get confused in the jump from connectivity graph to geometric mesh...

I hope the questions 1 and 2 are trivial and the methods I need for 3 and 4 are pretty standard.
Cheers, and thanks guys for your time!