How to use the edge routing of LayeredDigraphEmbedding when specifying explicit vertex coordinates?

Question

The "LayeredDigraphEmbedding" GraphLayout uses an edge routing algorithm which produces nice and readable diagrams and curved edges.

g = Graph[{1 -> 2, 2 -> 3, 1 -> 4, 4 -> 3, 4 -> 2, 4 -> 5}, 
  EdgeStyle -> Arrowheads[{{Medium, 0.5}}], VertexLabels -> "Name", 
  GraphLayout -> "LayeredDigraphEmbedding"]

How can I access this edge routing algorithm without also letting it position vertices? It seems that as soon as I fix vertex coordinates, it ceases working:

coord = GraphEmbedding[g];

SetProperty[g, VertexCoordinates -> coord]

Why do I need this? First I would like to plot an acyclic directed graph like the one above, then I would like to add additional edges (some of which introduce cycles) without disturbing either the layout of the vertices or the layout of the already existing edges. For example, I might want to add an additional edge 1 -> 3 to the above graph (without overlapping with 1 -> 2 -> 3), then yet another one from 5 -> 1 which makes it acyclic. All this time I want to keep the exiting vertex and edge locations untouched.

Answer 1

Update: A more convenient way than the original post is to generate a seperate graph with only the new edges and combine the GraphicsGroupBoxes of the two graphs:

ClearAll[graphAddF]
graphAddF = RawBoxes[With[{gg2 = Cases[ToBoxes[#2], 
      GraphicsGroupBox[x_] :> x[[1]], {0, Infinity}][[1]]}, 
     Replace[ToBoxes[#], GraphicsGroupBox[{x_, y_}] :> 
       GraphicsGroupBox[{{x, gg2}, y}], {0, Infinity}]]] &;

Examples:

g = Graph[{1 -> 2, 2 -> 3, 1 -> 4, 4 -> 3, 4 -> 2, 4 -> 5}, 
   EdgeStyle -> Arrowheads[{{Medium, 0.5}}], BaseStyle -> Thick,
   VertexLabels -> "Name", GraphLayout -> "LayeredDigraphEmbedding", 
   ImagePadding -> 10, ImageSize -> 300];

{newedgesa, newedgesb} = {{1 -> 3, 5 -> 1}, {1 -> 3, 5 -> 1, 3 -> 5}};
{curvaturesa, curvaturesb} = {{-0.5, 0}, {-0.5, 0, 0}};

g2 = Graph[newedgesa,  VertexCoordinates -> GraphEmbedding[g][[{1, 3, 5}]], 
   ImageSize -> 200, BaseStyle -> {Thick, Red}, 
   EdgeShapeFunction -> Thread[newedgesa -> (curvedArcF[{{Large, .75}}] /@ curvaturesa)]];
g3 = Graph[newedgesb,  VertexCoordinates -> GraphEmbedding[g][[{1, 3, 5}]], 
  ImageSize -> 200, BaseStyle -> {Thick, Red}, 
  EdgeShapeFunction -> Thread[newedgesb -> (curvedArcF[{{Large, .75}}] /@ curvaturesb)]]; 

Row[{g2, graphAddF[g, g2], g3, graphAddF[g, g3]}]

---

Original post:

A work-around until someone posts a more direct answer on the inner workings of edge routing in various embeddings:

to keep the existing vertex and edge locations untouched

We can construct EdgeShapeFunctions extracting the edge primitives from the box expression of a graph:

ClearAll[edgeRoutesF, curvedArcF]
edgeRoutesF[g_Graph] := Module[{grgrp = Cases[ToBoxes[g], 
   GraphicsGroupBox[x_] :> (x[[1]] /. {dirs___, sb : StyleBox[_, __] ..} :> 
    {StyleBox[#, ## & @@Flatten[{dirs, #2}]] & @@@ {sb}} /. 
    {DynamicLocation[v1_, ___], mid___, DynamicLocation[v2_, ___]} :> {v1, mid, v2}), 
  {0, Infinity}][[1, 1]], edges}, 
  edges = Cases[grgrp, {v1_String, mid___, v2_String} :> 
    (DirectedEdge @@ (ToExpression /@ StringSplit[{v1, v2}, "$"][[All, -1]])), 
    {0, Infinity}]; 
  Thread[edges -> (Function /@ (grgrp /. {v1_String, mid___, v2_String} :> 
    {#[[1]], mid, #[[2]]}))]]

To construct BezierCurves for the newly added edges we can use the built-in (but undocumented) EdgeShapeFunction "CurvedArc":

curvedArcF[ah_: {{Medium, .5}}][curv_: .5] := Composition[Style[#, Arrowheads[ah]] &, 
  Arrow, GraphElementData[{"CurvedArc", "Curvature" -> curv}]]

Examples:

g = Graph[{1 -> 2, 2 -> 3, 1 -> 4, 4 -> 3, 4 -> 2, 4 -> 5}, 
   EdgeStyle -> Arrowheads[{{Medium, 0.5}}], VertexLabels -> "Name", 
   GraphLayout -> "LayeredDigraphEmbedding", ImagePadding -> 10, ImageSize -> 300];

newedgesa = DirectedEdge @@@ {{1, 3}, {5, 1}};
curvaturesa = {-.5, -1.5};
edgeshapesa = Join[edgeRoutesF@g, 
   Thread[newedgesa -> (curvedArcF[{{Large, .75}}] /@ curvaturesa)]];

ga = SetProperty[EdgeAdd[g, newedgesa], 
   {EdgeShapeFunction -> edgeshapesa, VertexCoordinates -> GraphEmbedding[g],
    ImageSize -> 400}];

newedgesb = Join[newedgesa, {DirectedEdge[3, 5]}];
curvaturesb = Join[curvaturesa, {0}];
edgeshapesb = Join[edgeRoutesF@g, 
   Thread[newedgesb -> (curvedArcF[{{Large, .75}}] /@ curvaturesb)]];

gb = SetProperty[EdgeAdd[g, newedgesb], 
  {EdgeShapeFunction -> edgeshapesb, VertexCoordinates -> GraphEmbedding[g], 
  ImageSize -> 400}];

Row[{g, ga, gb}]

With GraphLayout -> "SpringElectricalEmbedding" and curvaturesa = {1.5, .5} we get

Row[{g, ga, gb}]