DepthFirstScan to work properly on Graph

Questions
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1

This is the raw data for the graph

matr1 = {{0, 0, 0, 0, 1, 1, 0}, {0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0,0, 1}, 
         {0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0}};

g = AdjacencyGraph[matr1, VertexLabels -> "Name", ImagePadding -> 10]

aada234g54trhj7654345trgfdf.jpg

That’s what gives me a built-in function. If you try a littl it’s easy to get all the paths which have 4 vertices.

Reap[DepthFirstScan[g, #, {"ForwardEdge" -> Sow, "BackEdge" -> Sow, 
       "FrontierEdge" -> Sow, "CrossEdge" -> Sow}]][[2]] & /@ Range@Length@matr1 // ColumnForm

Here is the 2nd answer that I got with my own function of in-depth-search (each path passes through 4 vertices, ie it consists of 3 edges):

{
 {1, 5, 1, 5},
 {1, 5, 1, 6},
 {1, 6, 1, 5},
 {1, 6, 1, 6},
 {1, 6, 2, 6},
 {2, 6, 1, 5},
 {2, 6, 1, 6},
 {2, 6, 2, 6},
 {3, 7, 3, 7},
 {5, 1, 5, 1},
 {5, 1, 6, 1},
 {5, 1, 6, 2},
 {6, 1, 5, 1},
 {6, 1, 6, 1},
 {6, 1, 6, 2},
 {6, 2, 6, 1},
 {6, 2, 6, 2},
 {7, 3, 7, 3}
}

Of course, I want DepthFirstScan to gave the same result. And I do not know how to make it.

Any advice?

Thanks,
Gregory

2

Vertices 3 and 7 are only connected to each other. I do not see how a depth-first scan would have a path of length four involving those vertices.
Danny

3

I got this answer (paths) using my own recursive function with the name FindAllPaths (a brilliant one, is it not?). And I just wonder if I can get the same result using built-in graph functions. In my function I have to introduce two additional vertices, a source and a sink, and then I search for all paths from the former to the latter.