image processing - How to generate a Graph from a picture of a graph?

I have a picture like this

I can get this graph's vertexes

mor = MorphologicalBinarize[img // ColorNegate] // 
     DeleteSmallComponents // ColorNegate // MorphologicalComponents;
selectmor = 
  SelectComponents[
    mor, {"AreaRadiusCoverage", "Count"}, # > 0.78 && #2 < 1000 &] // 

   Image;
mask = Graphics@
  Point@ComponentMeasurements[selectmor, "Centroid"][[All, 2]]

Or get its edges

fillimg = 
  FillingTransform[
     MorphologicalBinarize[img // ColorNegate] // 
        DeleteSmallComponents // Thinning // Pruning, 

     selectmor // Binarize] // Thinning // Pruning;
lines = ImageLines[fillimg, MaxFeatures -> 200, 
   Method -> {"Hough", "Segmented" -> True}];
Graphics[{Thick, Red, Line /@ lines}]

This my current work. May be you can realize it by more smart method. My target is not a Image but a Graph. So how to rebuild the Graph by this vertex and this edge?

---

Update: After the M.R.'s comments (Thanks for your comment), we can use the MorphologicalGraph to get the embryo of graph.

fillimg // MorphologicalGraph[#, VertexLabels -> "Name"] &

But it have a low precision. So How to refine it?

Answer

i       = Binarize@Import["http://i.stack.imgur.com/qofeF.png"];
vertexI = SelectComponents[i, "Count", 10 < # < 100 &];
vxPos   = ComponentMeasurements[vertexI, "Centroid"];
lines   = Subsets[Range@Length@vxPos, {2}];
linePos = lines /. vxPos;
ti@x_  := Total@Flatten@ImageData@Binarize@x


p = Position[ti@i - ti@Show[i, Graphics@Line@#] & /@ linePos, x_ /; x < 100];

Graphics[{Red, Line[Extract[linePos, p]]}]

Graph[Range@Length@vxPos, UndirectedEdge @@@ Extract[lines, p], 
      VertexLabels -> "Name", ImagePadding -> 20, 
      VertexCoordinates -> vxPos]