GraphUnion but with multiple edges?

Question

I have two graphs G1 and G2 with the same vertex set. I would like to take the union of G1 and G2 in such a way that if both G1 and G2 contain the same edge, then the union inherits both of those edges, labeled appropriately. Unfortunately, GraphUnion[G1, G2] seems to lump common edges into one. How can I accomplish my goal?

Answer 1

Update: If EdgeStyle and EdgeLabels for multi-edges are given as lists, we can use the option EdgeShapeFunction to label and style each edge individually:

graphSum = Module[{esf, styles = GroupBy[Flatten[PropertyValue[#, EdgeStyle] & /@ {##}], 
       First -> Last, Flatten],
     labels = GroupBy[Flatten[PropertyValue[#, EdgeLabels] & /@ {##}], 
       First -> Last, Flatten]},
    esf = {Dashing[{}], Text[First[labels[#2] = RotateRight[labels[#2] ]], 
        BSplineFunction[#][1/2]], 
       First[styles[#2] = RotateRight[styles[#2] ]], Arrow[#]} &;
    GraphComputation`GraphSum[##, VertexLabels -> "Name", EdgeShapeFunction -> esf]] &;

Examples:

ga = Graph[{1 -> 2}, EdgeStyle -> {_ :> Red}, 
   EdgeLabels -> {e_ :> Style[{e, "ga"}, 16, Black, Bold]}];
gb = Graph[{1 -> 2, 1 -> 2, 1 -> 2, 2 -> 3}, 
   EdgeStyle -> {_ :> Green , (1 -> 2) -> {Blue, Orange, Magenta}}, 
   EdgeLabels -> {e_ :> Style[{e, "gb"}, 16, Black, Bold], (1 -> 2) ->
      Array[Style[{#, DirectedEdge[1, 2], "gb"}, 16, Black, Bold]&, 3]}] ; 
SetProperty[graphSum[ga, gb], ImageSize -> 800]

SetProperty[graphSum[ga, ga, ga, gb, gb], 
 {ImageSize -> 800, VertexLabels -> Placed["Name", Center], 
  VertexShape -> Graphics[{EdgeForm[Gray], FaceForm[LightBlue], 
     Disk[{0, 0}, Offset[8]]}], AspectRatio -> 1/3}] 

Original answer:

GraphComputation`GraphSum gives the desired result:

g1 = Graph[{1 -> 2}];
g2 = Graph[{1 -> 2, 2 -> 3}];
GraphComputation`GraphSum[g1, g2, VertexLabels -> "Name", EdgeLabels -> "Name" ]

SeedRandom[1]
ga = RandomGraph[{5, 7}, DirectedEdges -> True, VertexLabels -> "Name", 
  ImageSize -> 300];
gb = RandomGraph[{10, 10},  DirectedEdges -> True, VertexLabels -> "Name",
  ImageSize -> 300];
Row[{ga, gb, HighlightGraph[GraphComputation`GraphSum[ga, gb, VertexLabels -> "Name", 
    ImageSize -> 500 ], {ga, gb}]}] 

Answer 2

If you have precisely the same vertices in precisely the same order (i.e. VertexList[g1] === VertexList[g2]), then you can just add the adjacency matrices.

g1 = Graph[{1, 2, 3}, {1 <-> 2}]

g2 = Graph[{1, 2, 3}, {1 <-> 2, 2 <-> 3}]

AdjacencyGraph[VertexList[g1], AdjacencyMatrix[g1] + AdjacencyMatrix[g2]]

Alternatively, join the edge lists. If you do this, it does not matter if the vertex lists are ordered the same. They just need to contain the same vertices.

Graph[VertexList[g1], Join[EdgeList[g1], EdgeList[g2]]]