Needs["PolyhedronOperations`"];
d = "Dodecahedron";
k = PolyhedronData[d, "SkeletonGraph"];
vc = PropertyValue[k, VertexCoordinates]
Style[Show[
Graphics[{BezierCurve[#, SplineDegree -> 2]}] & /@ ((vc[[#]] & /@
List @@@ EdgeList[k]) /. {x_, y_} -> {x, .3 Mean[{x, y}], y})],
Antialiasing -> True]