Orthogonal Graph layout

=== UPDATE ===

Functionality and concept are updated and discussed here:

Orthogonal aka rectangular edge layout for Graph

=== OLDER ===

The main problem here I think is laying out edges along orthogonal lines. This can be addressed with splines. First define function that triples every element in the list to make a spline to pass sharply through the points.

mlls[l_] := Flatten[Transpose[Table[l, {i, 3}]], 1];

In the function below I'll define a special EdgeRenderingFunction, a trick learned from @Yu-SungChang . Using LayeredGraphPlot:

OrthoLayer[x_] := LayeredGraphPlot[x, VertexLabeling -> True,
  PlotStyle -> Directive[Arrowheads[{{.02, .8}}], GrayLevel[.3]],

  EdgeRenderingFunction -> (Arrow@
      BezierCurve[
       mlls[{First[#1], {(1 First[#1][[1]] + 2 Last[#1][[1]])/3, 
          First[#1][[2]]}, {(1 First[#1][[1]] + 2 Last[#1][[1]])/3, 
          Last[#1][[2]]}, Last[#1]}]] &)]

Now I will use data from HERE and test the function

OrthoLayer[g]

Or similarly using Graph function:

OrthoLayer[x_] := Graph[x,
   GraphLayout -> "LayeredDrawing",
  VertexLabels -> "Name", VertexSize -> .1, VertexStyle -> Red,
  EdgeStyle -> Directive[Arrowheads[{{.015, .8}}], GrayLevel[.3]],
  EdgeShapeFunction -> (Arrow@
      BezierCurve[
       mlls[{First[#1], {(1 First[#1][[1]] + 2 Last[#1][[1]])/3, 
          First[#1][[2]]}, {(1 First[#1][[1]] + 2 Last[#1][[1]])/3, 
          Last[#1][[2]]}, Last[#1]}]] &), 
  PlotRange -> {{-.1, 4.4}, {-.1, 2.5}}]

OrthoLayer[g]

Using splines allows us to take advantage of various GraphLayout settings and still keep orthogonal edges.

g = {"John" -> "plants", "lion" -> "John", "tiger" -> "John", 
   "tiger" -> "deer", "lion" -> "deer", "deer" -> "plants", 
   "mosquito" -> "lion", "frog" -> "mosquito", "mosquito" -> "tiger", 
   "John" -> "cow", "cow" -> "plants", "mosquito" -> "deer", 
   "mosquito" -> "John", "snake" -> "frog", "vulture" -> "snake"};

OrthoLayer[x_, st_] := 
 Graph[x, GraphLayout -> st, VertexLabels -> "Name", VertexSize -> .3,
   VertexStyle -> Red, 
  EdgeStyle -> Directive[Arrowheads[{{.015, .8}}], GrayLevel[.3]], 
  EdgeShapeFunction -> (Arrow@
      BezierCurve[
       mlls[{First[#1], {(1 First[#1][[1]] + 2 Last[#1][[1]])/3, 
          First[#1][[2]]}, {(1 First[#1][[1]] + 2 Last[#1][[1]])/3, 
          Last[#1][[2]]}, Last[#1]}]] &), PlotRange -> All, 
  PlotRangePadding -> .2]

OrthoLayer[g, #] & /@ {"CircularEmbedding", "LayeredDrawing", 
  "RandomEmbedding", "SpiralEmbedding", "SpringElectricalEmbedding", 
  "SpringEmbedding"}

Not perfect, but a start. Many things can be adjusted to customize specific data.

Consider a sample graph.

g = RandomGraph[BernoulliGraphDistribution[10, 0.5]]

We can get the actual vertex list like this:

PropertyValue[{g, #}, VertexCoordinates] & /@ VertexList[g]
(*  {{2.74373, 0.537705}, {1.34615, 1.02584}, {1.82543, 1.22826}, {0., 
   0.531856}, {0.755493, 0.678809}, {1.4028, 2.11914}, {0.991734, 
   0.}, {1.74247, 0.190579}, {0.84817, 1.39096}, {1.94835, 0.6529}} *)

We can shift those vertices onto a grid by rounding the coordinates in a suitable way (note Round is Listable):

ongrid = Round[
   PropertyValue[{g, #}, VertexCoordinates] & /@ VertexList[g], 1/2]

We can then apply these new coordinate rules to the original graph through a slightly convoluted use of SetProperty.

g2 = Fold[SetProperty[{#1, #2[[1]]}, VertexCoordinates -> #2[[2]]] &, 
   g, Transpose[{VertexList[g], ongrid}] ]

This can all be bound up in a custom function like so:

Clear[LayoutOnGrid]

LayoutOnGrid[g_Graph, d_?NumericQ] :=
 Module[{v = VertexList[g], grid}, 
  grid = Round[
    PropertyValue[{g, #}, VertexCoordinates] & /@ v, d];
  Fold[SetProperty[{#1, #2[[1]]}, VertexCoordinates -> #2[[2]]] &, g, 
   Transpose[{v, grid}] ]]

Note that you might need to tweak how to round the coordinate locations to get a nice look. Rounding to 1 tends to put lots of vertices on top of each other.

LayoutOnGrid[RandomGraph[BernoulliGraphDistribution[20, 0.5]], 1/2]