Depending on your hardware, this function may be useful
manipulatablegraph[graph_Graph] :=
DynamicModule[{verts, vertcoords, edges, edgestates, edgehandlers},
verts = VertexList@graph;
vertcoords = AbsoluteOptions[graph, VertexCoordinates];
edges = EdgeList@graph; edgestates = Table[1, {e, edges}];
edgehandlers =
MapIndexed[
EventHandler[#1, {"MouseClicked" :> (edgestates[[First@#2]] =
Mod[edgestates[[First@#2]] + 1, 2])}] &, edges];
Column@{Button["export selected edges",
Print[InputForm@
Graph[verts,
edges[[Select[Range@Length@edges, edgestates[[#]] == 0 &]]],
vertcoords]]],
Dynamic@Graph[verts, edgehandlers, vertcoords,
EdgeStyle ->
Table[edges[[i]] -> {Thickness[.015],
If[edgestates[[i]] == 0, Blue, Lighter[Blue, .9]]}, {i,
Length@edges}]]}]
(*borrows heavily from https://mathematica.stackexchange.com/questions/51241/\
generating-eventhandler-using-table*)
Call it on your graph. You should be able to click edges to toggle them from light to solid blue. When you press 'export selected edges', it prints out the code for a graph whose edges are currently solid (intended for copying and pasting as Mathematica input).
If you don't insist on that particular edge subset (and don't want to select edges manually), you can coax Mathematica to output a spanning tree with a specified root node by iteratively calling FindSpanningTree and removing key edges.
I imagine Illustrator can export to many file formats. One of these might be easy to parse with Mathematica. There's always the nasty method of doing image processing on your result, and constructing graphs from morphology (it's surprisingly simple and robust).
