Combining two directed graphs without loosing any edge

Question

I have two directed graphs given below:

one={1 -> 2, 1 -> 3, 3 -> 4, 4 -> 1};
two={1 -> 2, 1 -> 3, 1 -> 4};

I simply want to map the two graphs as a single graph keeping all the edges repeated as multiple edges like 1->2, 1->3. I tried

GraphUnion[one, two]

but it is not giving me what I am after. I want to keep all the edges in these two graphs and differentiate the two graphs with different colors.

Any idea?

Example 2

one = {1 \[DirectedEdge] 10, 3 \[DirectedEdge] 5, 3 \[DirectedEdge] 6,
6 \[DirectedEdge] 1, 6 \[DirectedEdge] 2, 6 \[DirectedEdge] 3, 
6 \[DirectedEdge] 4, 6 \[DirectedEdge] 7, 6 \[DirectedEdge] 8, 
6 \[DirectedEdge] 11, 6 \[DirectedEdge] 12, 6 \[DirectedEdge] 13, 
6 \[DirectedEdge] 14, 6 \[DirectedEdge] 15, 6 \[DirectedEdge] 16, 
6 \[DirectedEdge] 17, 11 \[DirectedEdge] 9};    

two = {1 \[DirectedEdge] 6, 2 \[DirectedEdge] 6, 3 \[DirectedEdge] 6, 
4 \[DirectedEdge] 6, 5 \[DirectedEdge] 6, 6 \[DirectedEdge] 8, 
7 \[DirectedEdge] 6, 8 \[DirectedEdge] 6, 9 \[DirectedEdge] 8, 
10 \[DirectedEdge] 6, 11 \[DirectedEdge] 9, 12 \[DirectedEdge] 6, 
13 \[DirectedEdge] 9, 14 \[DirectedEdge] 6, 15 \[DirectedEdge] 6, 
16 \[DirectedEdge] 6, 17 \[DirectedEdge] 6};

HighlightGraph[EdgeAdd[Graph[one, VertexLabels -> "Name"], two], two, 
VertexLabels -> "Name"]
GraphUnion[one, two, VertexLabels -> "Name"]

In Highlight there is something wrong because there are two edges like 11->9, and only one of them should be RED, the other be BLUE. It seems that identical edges are assumed to belong to graph two only. What I am after is to separate them with different colors as the colors have a meaning in the context. GraphUnion is not generating the same graph at all because it takes mathematical union of the two graphs, eliminating the repeated edges.

I simply want to map two separate graphs with different colors on top of each other without loosing any directed edge.

Answer 1

You can use EdgeAdd:

EdgeAdd[one, two, VertexLabels -> "Name"]

Alternatively, you can use GraphComputation`GraphSum:

GraphComputation`GraphSum[Graph @ one, Graph @ two, VertexLabels->"Name"]

same picture

GraphUnion works as expected (that is, it will not the give desired output): it takes the Union of vertices and the Union of edges from the two input graphs:

GraphUnion -> Details and Options:

• The graph union Graph[$v_1, e_1$] ⋃ Graph[$v_2, e_2$] is given by Graph[$v_1$ ⋃ $v_2$, $e_1$ ⋃ $e_2$].