Algorithm to find all the cycle bases in a graph I am given a graph defined by vertexes and edges. I have to obtain all the cycle bases in a network. No coordinates will be given for the nodes.
Here's a sketch that illustrates my point.
Note that inside a cycle it must not contain any edge
 A: According to your sketch, you don't want cycles that enclose more than one face Ri. For instance, you are not interested in the cycle 9-8-14-16-15-13, because it contains two faces, R4 and R5.
But if this is the case, your problem is ill-posed. If I don't have the coordinates of the nodes, then for all I know, nodes 15 and 16 could lie on the other side of L14, inside R4. And in this case, I would want to include the cycle 9-8-14-16-15-13.
So you have to decide: do you give me the coordinates? Or do you want every cycle in the graph, including for instance 2-10-9-8-14-16-15-13-12-11?
A: It seems like the OP is looking for a list of faces and their boundaries for a planar graph.  However, without coordinates or an embedding in the plane, this is definitely ill-posed.  As a simple counterexample, consider the complete graph K4.  This has 4 possible faces (123,124,134,234), but any embedding in the plane has only 3 of them.  This leads to 4 different possible answers, for the same graph, depending on which vertex is placed in the center of the other three.  This means that, without more information, the problem doesn't have a unique answer.
A: Maybe what you want is a cycle basis? That is, a set of cycles such that any other cycle can be found by adding and subtracting combinations of cycles in the basis. One can find a cycle basis easily for any graph by finding a spanning tree and then, for each edge that's not in the tree, reporting the cycle formed by that edge together with the tree path connecting its endpoints. In a plane-embedded graph, the set of interior faces forms a cycle basis, matching what the sketch describes. Finding the shortest cycle basis is more complicated but still known in polynomial time; see e.g. Kavitha et al, ICALP 2004.
A: 
Note that inside a cycle it must not contain a link, all the cycle must be clean (no link) and closed.

Sigh This edit doesn't help the situation at all.
What does 'inside' mean, if your graph doesn't come with an embedding in the plane? How do I know whether nodes 15 and 16 lie 'inside' R4, if I don't have their coordinates? Look at Ari's answer for more enlightenment.
A: By 'clean cycle' do you perhaps mean 'chordless cycle'? This I think is well-defined without an embedding, as it's just a condition on adjacency of vertices. If so, this page seems to describe an algorithm for enumerating such cycles.
