diff --git a/src/cycles/basis.jl b/src/cycles/basis.jl
index f4d26a769..e26692384 100644
--- a/src/cycles/basis.jl
+++ b/src/cycles/basis.jl
@@ -29,7 +29,7 @@ julia> cycle_basis(g)
 ### References
 * Paton, K. An algorithm for finding a fundamental set of cycles of a graph. Comm. ACM 12, 9 (Sept 1969), 514-518. [https://dl.acm.org/citation.cfm?id=363232]
 """
-function cycle_basis(g::AbstractSimpleGraph, root=nothing)
+function cycle_basis(g::AbstractGraph, root=nothing)
     T = eltype(g)
     cycles = Vector{Vector{T}}()
 
diff --git a/src/cycles/johnson.jl b/src/cycles/johnson.jl
index 6f0d34983..52892942e 100644
--- a/src/cycles/johnson.jl
+++ b/src/cycles/johnson.jl
@@ -1,3 +1,7 @@
+# TODO most functions here do not work for general abstract graphs yet
+# as most of them relay on induced_subgraph, which expects the graph type
+# to be modifiable.
+
 abstract type Visitor{T<:Integer} end
 
 """
diff --git a/test/cycles/basis.jl b/test/cycles/basis.jl
index 7d17e3f8c..804aa1afd 100644
--- a/test/cycles/basis.jl
+++ b/test/cycles/basis.jl
@@ -8,7 +8,7 @@
     # No Edges
     ex = Graph(1)
     expected_cyclebasis = Array{Int64,1}[]
-    @testset "no edges" for g in testgraphs(ex)
+    @testset "no edges" for g in test_generic_graphs(ex)
         ex_cyclebasis = @inferred cycle_basis(g)
         @test isempty(ex_cyclebasis)
     end
@@ -17,7 +17,7 @@
     elist = [(1, 1)]
     ex = Graph(SimpleEdge.(elist))
     expected_cyclebasis = Array{Int64,1}[[1]]
-    @testset "one self-edge" for g in testgraphs(ex)
+    @testset "one self-edge" for g in test_generic_graphs(ex)
         ex_cyclebasis = cycle_basis(g)
         evaluate(ex_cyclebasis, expected_cyclebasis)
     end
@@ -26,7 +26,7 @@
     elist = [(1, 2), (2, 3), (3, 4), (4, 1), (1, 5)]
     ex = Graph(SimpleEdge.(elist))
     expected_cyclebasis = Array{Int64,1}[[1, 2, 3, 4]]
-    @testset "one cycle" for g in testgraphs(ex)
+    @testset "one cycle" for g in test_generic_graphs(ex)
         ex_cyclebasis = cycle_basis(g)
         evaluate(ex_cyclebasis, expected_cyclebasis)
     end
@@ -35,7 +35,7 @@
     elist = [(1, 2), (1, 3), (2, 3), (2, 4), (3, 4)]
     ex = Graph(SimpleEdge.(elist))
     expected_cyclebasis = Array{Int64,1}[[2, 3, 4], [2, 1, 3]]
-    @testset "2 of 3 cycles w/ basis" for g in testgraphs(ex)
+    @testset "2 of 3 cycles w/ basis" for g in test_generic_graphs(ex)
         ex_cyclebasis = cycle_basis(g)
         evaluate(ex_cyclebasis, expected_cyclebasis)
     end
@@ -44,7 +44,7 @@
     elist = [(1, 2), (1, 3), (2, 3), (2, 4), (3, 4), (1, 5), (5, 6), (6, 4)]
     ex = Graph(SimpleEdge.(elist))
     expected_cyclebasis = Array{Int64,1}[[2, 4, 3], [1, 5, 6, 4, 3], [1, 2, 3]]
-    @testset "root argument" for g in testgraphs(ex)
+    @testset "root argument" for g in test_generic_graphs(ex)
         ex_cyclebasis = @inferred cycle_basis(g, 3)
         evaluate(ex_cyclebasis, expected_cyclebasis)
     end
@@ -52,7 +52,7 @@
     @testset "two isolated cycles" begin
         ex = blockdiag(cycle_graph(3), cycle_graph(4))
         expected_cyclebasis = [[1, 2, 3], [4, 5, 6, 7]]
-        for g in testgraphs(ex)
+        for g in test_generic_graphs(ex)
             found_cyclebasis = @inferred cycle_basis(g)
             evaluate(expected_cyclebasis, found_cyclebasis)
         end
diff --git a/test/cycles/hawick-james.jl b/test/cycles/hawick-james.jl
index 636298f18..9cf987b51 100644
--- a/test/cycles/hawick-james.jl
+++ b/test/cycles/hawick-james.jl
@@ -11,7 +11,7 @@
 
     @testset "subset" for g in testgraphs(ex1)
         expected_circuits = Vector{Int}[[2, 3, 4, 5], [2, 3, 5]]
-        ex1_circuits = simplecycles_hawick_james(g)
+        ex1_circuits = simplecycles_hawick_james(GenericDiGraph(g))
 
         @test issubset(expected_circuits, ex1_circuits)
         @test issubset(ex1_circuits, expected_circuits)
@@ -20,7 +20,7 @@
         add_edge!(g, 1, 1)
         add_edge!(g, 3, 3)
 
-        ex1_circuits_self = simplecycles_hawick_james(g)
+        ex1_circuits_self = simplecycles_hawick_james(GenericDiGraph(g))
 
         @test issubset(expected_circuits, ex1_circuits_self)
         @test [1] ∈ ex1_circuits_self && [3] ∈ ex1_circuits_self
@@ -28,14 +28,14 @@
 
     # Path DiGraph
     ex2_size = 10
-    ex2 = testgraphs(path_digraph(ex2_size))
+    ex2 = test_generic_graphs(path_digraph(ex2_size))
     @testset "empty" for g in ex2
         @test isempty(simplecycles_hawick_james(g))
     end
 
     # Complete DiGraph
     ex3_size = 5
-    ex3 = testgraphs(complete_digraph(ex3_size))
+    ex3 = test_generic_graphs(complete_digraph(ex3_size))
     @testset "length" for g in ex3
         ex3_circuits = simplecycles_hawick_james(g)
         @test length(ex3_circuits) == length(unique(ex3_circuits))
@@ -62,7 +62,7 @@
         add_edge!(ex4, src, dest)
         add_edge!(ex4, dest, src)
     end
-    @testset "membership" for g in testgraphs(ex4)
+    @testset "membership" for g in test_generic_graphs(ex4)
         ex4_output = simplecycles_hawick_james(g)
         @test [1, 2] ∈ ex4_output && [8, 9] ∈ ex4_output
     end
@@ -72,8 +72,9 @@
         (n, k) in [(14, 18), (10, 22), (7, 16)]
 
         g = erdos_renyi(n, k; is_directed=true, rng=StableRNG(seed))
+        # TODO simplecycles(g) does not yet work with GenericDiGraph
         cycles1 = simplecycles(g)
-        cycles2 = simplecycles_hawick_james(g)
+        cycles2 = simplecycles_hawick_james(GenericDiGraph(g))
         foreach(sort!, cycles1)
         foreach(sort!, cycles2)
         sort!(cycles1)
diff --git a/test/cycles/incremental.jl b/test/cycles/incremental.jl
index c20bc815b..f8bd5b37d 100644
--- a/test/cycles/incremental.jl
+++ b/test/cycles/incremental.jl
@@ -1,3 +1,7 @@
+# The test here cannot be done with GenericDiGraph, as the functions under test
+# modify the graph. We probably need another generic graph type with more relaxed
+# constraints for modifying graphs.
+
 @testset "ICT" begin
     Gempty = SimpleDiGraph(3)
     Gsomedges = SimpleDiGraph(6)
diff --git a/test/cycles/johnson.jl b/test/cycles/johnson.jl
index aff6666ab..5e7a2882f 100644
--- a/test/cycles/johnson.jl
+++ b/test/cycles/johnson.jl
@@ -1,3 +1,6 @@
+# TODO we cannot run the test here with GenericGraph yet,
+# as the functions don't work for arbitrary graphs yet.
+
 @testset "Cycles" begin
     rng = StableRNG(1)
     completedg = complete_digraph(4)
diff --git a/test/cycles/karp.jl b/test/cycles/karp.jl
index a45324f27..faabac923 100644
--- a/test/cycles/karp.jl
+++ b/test/cycles/karp.jl
@@ -46,7 +46,7 @@
     add_edge!(g1, 11, 7)
     add_edge!(g1, 12, 9)
 
-    @testset "simple digraphs" for g in testgraphs(g1)
+    @testset "simple digraphs" for g in test_generic_graphs(g1)
         c, λ = karp_minimum_cycle_mean(g, w)
         @test c == [9, 11, 7]
         @test λ == 0.9
@@ -78,7 +78,7 @@
         1.0 Inf 0.0 Inf
     ]
 
-    @testset "tricky case" for g in testgraphs(tricky)
+    @testset "tricky case" for g in test_generic_graphs(tricky)
         c, λ = karp_minimum_cycle_mean(g, distmx)
         @test λ == 0.0
         @test sort(c) == [3, 4]
@@ -97,7 +97,7 @@
         Inf -1
     ]
 
-    @testset "multiple SCCs" for g in testgraphs(multi)
+    @testset "multiple SCCs" for g in test_generic_graphs(multi)
         c, λ = karp_minimum_cycle_mean(g, distmx)
         @test λ == -1.0
         @test c == [2]
diff --git a/test/cycles/limited_length.jl b/test/cycles/limited_length.jl
index f9c529033..051c06671 100644
--- a/test/cycles/limited_length.jl
+++ b/test/cycles/limited_length.jl
@@ -3,7 +3,7 @@
     pathdg = path_digraph(5)
     cycledg = cycle_digraph(5)
 
-    @testset "complete digraph" for g in testgraphs(completedg)
+    @testset "complete digraph" for g in test_generic_graphs(completedg)
         @test length(simplecycles_limited_length(g, 0)) == 0
         @test length(simplecycles_limited_length(g, 1)) == 0
         @test length(simplecycles_limited_length(g, 2)) == 6
@@ -13,7 +13,7 @@
         @test length(simplecycles_limited_length(g, 4, typemax(Int))) == 20
     end
 
-    @testset "path digraph" for g in testgraphs(pathdg)
+    @testset "path digraph" for g in test_generic_graphs(pathdg)
         @test length(simplecycles_limited_length(g, 1)) == 0
         @test length(simplecycles_limited_length(g, 2)) == 0
         @test length(simplecycles_limited_length(g, 3)) == 0
@@ -21,7 +21,7 @@
         @test length(simplecycles_limited_length(g, 5)) == 0
     end
 
-    @testset "cycle digraph" for g in testgraphs(cycledg)
+    @testset "cycle digraph" for g in test_generic_graphs(cycledg)
         @test length(simplecycles_limited_length(g, 1)) == 0
         @test length(simplecycles_limited_length(g, 2)) == 0
         @test length(simplecycles_limited_length(g, 3)) == 0
@@ -30,12 +30,12 @@
     end
 
     @testset "self loops" begin
-        selfloopg = DiGraph([
+        selfloopg = GenericDiGraph(DiGraph([
             0 1 0 0
             0 0 1 0
             1 0 1 0
             0 0 0 1
-        ])
+        ]))
         cycles = simplecycles_limited_length(selfloopg, nv(selfloopg))
         @test [3] in cycles
         @test [4] in cycles
@@ -46,10 +46,12 @@
     @testset "octahedral graph" begin
         octag = smallgraph(:octahedral)
         octadg = DiGraph(octag)
+        # TODO simplecycleslength does currently not yet work with GenericDiGraph.
+        # This is probably because it uses induced_subgraph which fails on that graph type.
         octalengths, _ = simplecycleslength(octadg)
         for k in 1:6
-            @test sum(octalengths[1:k]) == length(simplecycles_limited_length(octag, k))
-            @test sum(octalengths[1:k]) == length(simplecycles_limited_length(octadg, k))
+            @test sum(octalengths[1:k]) == length(simplecycles_limited_length(GenericGraph(octag), k))
+            @test sum(octalengths[1:k]) == length(simplecycles_limited_length(GenericDiGraph(octadg), k))
         end
     end
 end