diff --git a/nx_cugraph/classes/digraph.py b/nx_cugraph/classes/digraph.py
index 178bf44f1..a73ca8d0b 100644
--- a/nx_cugraph/classes/digraph.py
+++ b/nx_cugraph/classes/digraph.py
@@ -34,6 +34,7 @@
 __all__ = ["CudaDiGraph", "DiGraph"]
 
 networkx_api = nxcg.utils.decorators.networkx_class(nx.DiGraph)
+gpu_cpu_api = nxcg.utils.decorators._gpu_cpu_api(nx.DiGraph, __name__)
 
 
 class DiGraph(nx.DiGraph, Graph):
@@ -105,6 +106,43 @@ def to_cudagraph_class(cls) -> type[CudaDiGraph]:
     def to_networkx_class(cls) -> type[nx.DiGraph]:
         return nx.DiGraph
 
+    ##########################
+    # Networkx graph methods #
+    ##########################
+
+    # Dispatch to nx.DiGraph or CudaDiGraph
+    __contains__ = gpu_cpu_api("__contains__")
+    __len__ = gpu_cpu_api("__len__")
+    __iter__ = gpu_cpu_api("__iter__")
+
+    @networkx_api
+    def clear(self) -> None:
+        cudagraph = self._cudagraph if self._is_on_gpu else None
+        if self._is_on_cpu:
+            super().clear()
+        if cudagraph is not None:
+            cudagraph.clear()
+            self._set_cudagraph(cudagraph, clear_cpu=False)
+
+    @networkx_api
+    def clear_edges(self) -> None:
+        cudagraph = self._cudagraph if self._is_on_gpu else None
+        if self._is_on_cpu:
+            super().clear_edges()
+        if cudagraph is not None:
+            cudagraph.clear_edges()
+            self._set_cudagraph(cudagraph, clear_cpu=False)
+
+    get_edge_data = gpu_cpu_api("get_edge_data", edge_data=True)
+    has_edge = gpu_cpu_api("has_edge")
+    neighbors = gpu_cpu_api("neighbors")
+    has_node = gpu_cpu_api("has_node")
+    nbunch_iter = gpu_cpu_api("nbunch_iter")
+    number_of_edges = Graph.number_of_edges
+    number_of_nodes = gpu_cpu_api("number_of_nodes")
+    order = gpu_cpu_api("order")
+    successors = gpu_cpu_api("successors")
+
 
 class CudaDiGraph(CudaGraph):
     #################
@@ -244,6 +282,7 @@ def to_undirected(self, reciprocal=False, as_view=False):
             rv.graph.update(deepcopy(self.graph))
         return rv
 
+    successors = CudaGraph.neighbors  # Alias
     # Many more methods to implement...
 
     ###################
diff --git a/nx_cugraph/classes/graph.py b/nx_cugraph/classes/graph.py
index 11c6215d8..018636228 100644
--- a/nx_cugraph/classes/graph.py
+++ b/nx_cugraph/classes/graph.py
@@ -48,6 +48,7 @@
 __all__ = ["CudaGraph", "Graph"]
 
 networkx_api = nxcg.utils.decorators.networkx_class(nx.Graph)
+gpu_cpu_api = nxcg.utils.decorators._gpu_cpu_api(nx.Graph, __name__)
 
 # The "everything" cache key is an internal implementation detail of NetworkX
 # that may change between releases.
@@ -63,6 +64,8 @@
         True,  # Include all node values
         # `.graph` attributes are always included now
     )
+_EDGE_KEY_INDEX = 0
+_NODE_KEY_INDEX = 1
 
 # Use to indicate when a full conversion to GPU failed so we don't try again.
 _CANT_CONVERT_TO_GPU = "_CANT_CONVERT_TO_GPU"
@@ -210,8 +213,7 @@ def _cudagraph(self):
         cache[_CACHE_KEY] = Gcg
         return Gcg
 
-    @_cudagraph.setter
-    def _cudagraph(self, val, *, clear_cpu=True):
+    def _set_cudagraph(self, val, *, clear_cpu=True):
         """Set the full ``CudaGraph`` for this graph, or remove from device if None."""
         if (cache := getattr(self, "__networkx_cache__", None)) is None:
             # Should we warn?
@@ -228,6 +230,32 @@ def _cudagraph(self, val, *, clear_cpu=True):
                 for key in self._nx_attrs:
                     self.__dict__[key] = None
 
+    def _get_cudagraph(self, *, edge_data=False, node_data=False):
+        """Get a valid cached ``CudaGraph``, optionally with edge or node data.
+
+        Returns None if no valid graph is found.
+
+        Parameters
+        ----------
+        edge_data : bool, default False
+            Whether to return a CudaGraph with edge data.
+        node_data : bool, default False
+            Whether to return a CudaGraph with node data.
+        """
+        nx_cache = getattr(self, "__networkx_cache__", None)
+        if nx_cache is None or _CANT_CONVERT_TO_GPU in nx_cache:
+            return None
+        cache = nx_cache.get("backends", {}).get("cugraph", {})
+        if _CACHE_KEY in cache:
+            # Always return the canonical CudaGraph if it exists
+            return cache[_CACHE_KEY]
+        for key, val in cache.items():
+            if (key[_EDGE_KEY_INDEX] is True or edge_data is False) and (
+                key[_NODE_KEY_INDEX] is True or node_data is False
+            ):
+                return val
+        return None
+
     @nx.Graph.name.setter
     def name(self, s):
         # Don't clear the cache when setting the name, since `.graph` is shared.
@@ -509,6 +537,54 @@ def from_dcsc(
             **attr,
         )
 
+    ##########################
+    # Networkx graph methods #
+    ##########################
+
+    # Dispatch to nx.Graph or CudaGraph
+    __contains__ = gpu_cpu_api("__contains__")
+    __len__ = gpu_cpu_api("__len__")
+    __iter__ = gpu_cpu_api("__iter__")
+
+    @networkx_api
+    def clear(self) -> None:
+        cudagraph = self._cudagraph if self._is_on_gpu else None
+        if self._is_on_cpu:
+            super().clear()
+        if cudagraph is not None:
+            cudagraph.clear()
+            self._set_cudagraph(cudagraph, clear_cpu=False)
+
+    @networkx_api
+    def clear_edges(self) -> None:
+        cudagraph = self._cudagraph if self._is_on_gpu else None
+        if self._is_on_cpu:
+            super().clear_edges()
+        if cudagraph is not None:
+            cudagraph.clear_edges()
+            self._set_cudagraph(cudagraph, clear_cpu=False)
+
+    get_edge_data = gpu_cpu_api("get_edge_data", edge_data=True)
+    has_edge = gpu_cpu_api("has_edge")
+    neighbors = gpu_cpu_api("neighbors")
+    has_node = gpu_cpu_api("has_node")
+    nbunch_iter = gpu_cpu_api("nbunch_iter")
+
+    @networkx_api
+    def number_of_edges(
+        self, u: NodeKey | None = None, v: NodeKey | None = None
+    ) -> int:
+        if u is not None or v is not None:
+            # NotImplemented by CudaGraph
+            nx_class = self.to_networkx_class()
+            return nx_class.number_of_edges(self, u, v)
+        return self._number_of_edges(u, v)
+
+    _number_of_edges = gpu_cpu_api("number_of_edges")
+    number_of_nodes = gpu_cpu_api("number_of_nodes")
+    order = gpu_cpu_api("order")
+    # Future work: implement more graph methods, and handle e.g. `copy`
+
 
 class CudaGraph:
     # Tell networkx to dispatch calls with this object to nx-cugraph
@@ -803,7 +879,7 @@ def to_undirected(self, as_view: bool = False) -> CudaGraph:
 
     def _to_compat_graph(self) -> Graph:
         rv = self._to_compat_graph_class()()
-        rv._cudagraph = self
+        rv._set_cudagraph(self)
         return rv
 
     # Not implemented...
diff --git a/nx_cugraph/classes/multidigraph.py b/nx_cugraph/classes/multidigraph.py
index 5a6595567..0671d21a8 100644
--- a/nx_cugraph/classes/multidigraph.py
+++ b/nx_cugraph/classes/multidigraph.py
@@ -23,6 +23,7 @@
 __all__ = ["CudaMultiDiGraph", "MultiDiGraph"]
 
 networkx_api = nxcg.utils.decorators.networkx_class(nx.MultiDiGraph)
+gpu_cpu_api = nxcg.utils.decorators._gpu_cpu_api(nx.MultiDiGraph, __name__)
 
 
 class MultiDiGraph(nx.MultiDiGraph, MultiGraph, DiGraph):
@@ -50,6 +51,26 @@ def to_cudagraph_class(cls) -> type[CudaMultiDiGraph]:
     def to_networkx_class(cls) -> type[nx.MultiDiGraph]:
         return nx.MultiDiGraph
 
+    ##########################
+    # Networkx graph methods #
+    ##########################
+
+    # Dispatch to nx.MultiDiGraph or CudaMultiDiGraph
+    __contains__ = gpu_cpu_api("__contains__")
+    __len__ = gpu_cpu_api("__len__")
+    __iter__ = gpu_cpu_api("__iter__")
+    clear = DiGraph.clear
+    clear_edges = DiGraph.clear_edges
+    get_edge_data = gpu_cpu_api("get_edge_data", edge_data=True)
+    has_edge = gpu_cpu_api("has_edge")
+    neighbors = gpu_cpu_api("neighbors")
+    has_node = gpu_cpu_api("has_node")
+    nbunch_iter = gpu_cpu_api("nbunch_iter")
+    number_of_edges = MultiGraph.number_of_edges
+    number_of_nodes = gpu_cpu_api("number_of_nodes")
+    order = gpu_cpu_api("order")
+    successors = gpu_cpu_api("successors")
+
 
 class CudaMultiDiGraph(CudaMultiGraph, CudaDiGraph):
     is_directed = classmethod(MultiDiGraph.is_directed.__func__)
diff --git a/nx_cugraph/classes/multigraph.py b/nx_cugraph/classes/multigraph.py
index c8c8f1dfb..3f0204f69 100644
--- a/nx_cugraph/classes/multigraph.py
+++ b/nx_cugraph/classes/multigraph.py
@@ -37,6 +37,7 @@
 __all__ = ["MultiGraph", "CudaMultiGraph"]
 
 networkx_api = nxcg.utils.decorators.networkx_class(nx.MultiGraph)
+gpu_cpu_api = nxcg.utils.decorators._gpu_cpu_api(nx.MultiGraph, __name__)
 
 
 class MultiGraph(nx.MultiGraph, Graph):
@@ -277,6 +278,36 @@ def from_dcsc(
             **attr,
         )
 
+    ##########################
+    # Networkx graph methods #
+    ##########################
+
+    # Dispatch to nx.MultiGraph or CudaMultiGraph
+    __contains__ = gpu_cpu_api("__contains__")
+    __len__ = gpu_cpu_api("__len__")
+    __iter__ = gpu_cpu_api("__iter__")
+    clear = Graph.clear
+    clear_edges = Graph.clear_edges
+    get_edge_data = gpu_cpu_api("get_edge_data", edge_data=True)
+    has_edge = gpu_cpu_api("has_edge")
+    neighbors = gpu_cpu_api("neighbors")
+    has_node = gpu_cpu_api("has_node")
+    nbunch_iter = gpu_cpu_api("nbunch_iter")
+
+    @networkx_api
+    def number_of_edges(
+        self, u: NodeKey | None = None, v: NodeKey | None = None
+    ) -> int:
+        if u is not None or v is not None:
+            # NotImplemented by CudaGraph
+            nx_class = self.to_networkx_class()
+            return nx_class.number_of_edges(self, u, v)
+        return self._number_of_edges(u, v)
+
+    _number_of_edges = gpu_cpu_api("number_of_edges")
+    number_of_nodes = gpu_cpu_api("number_of_nodes")
+    order = gpu_cpu_api("order")
+
 
 class CudaMultiGraph(CudaGraph):
     # networkx properties
@@ -390,14 +421,13 @@ def get_edge_data(
         mask = (self.src_indices == u) & (self.dst_indices == v)
         if not mask.any():
             return default
-        if self.edge_keys is None:
+        if self.edge_keys is None and key is not None:
             if self.edge_indices is None:
                 self._calculate_edge_indices()
-            if key is not None:
-                try:
-                    mask = mask & (self.edge_indices == key)
-                except TypeError:
-                    return default
+            try:
+                mask = mask & (self.edge_indices == key)
+            except TypeError:
+                return default
         indices = cp.nonzero(mask)[0]
         if indices.size == 0:
             return default
diff --git a/nx_cugraph/tests/test_graph_methods.py b/nx_cugraph/tests/test_graph_methods.py
index 40a361b10..abc71ed77 100644
--- a/nx_cugraph/tests/test_graph_methods.py
+++ b/nx_cugraph/tests/test_graph_methods.py
@@ -17,6 +17,8 @@
 
 from .testing_utils import assert_graphs_equal
 
+CREATE_USING = [nxcg.Graph, nxcg.DiGraph, nxcg.MultiGraph, nxcg.MultiDiGraph]
+
 
 def _create_Gs():
     rv = []
@@ -65,3 +67,51 @@ def test_multidigraph_to_undirected():
     Gcg = nxcg.CudaMultiDiGraph(Gnx)
     with pytest.raises(NotImplementedError):
         Gcg.to_undirected()
+
+
+@pytest.mark.parametrize("create_using", CREATE_USING)
+@pytest.mark.parametrize(
+    "method",
+    [
+        ("__iter__", ()),
+        ("__len__", ()),
+        ("clear", ()),
+        ("clear_edges", ()),
+        ("nbunch_iter", ()),
+        ("number_of_edges", ()),
+        ("number_of_nodes", ()),
+        ("order", ()),
+        ("__contains__", (0,)),
+        ("neighbors", (0,)),
+        ("has_node", (0,)),
+        ("successors", (0,)),
+        ("get_edge_data", (0, 1)),
+        ("has_edge", (0, 1)),
+        ("nbunch_iter", ([0, 1],)),
+    ],
+)
+@pytest.mark.parametrize("where", ["gpu", "cpu"])
+def test_method_does_not_convert_to_cpu_or_gpu(create_using, method, where):
+    attr, args = method
+    if attr == "successors" and not create_using.is_directed():
+        return
+    G = nxcg.complete_graph(3, create_using=create_using)
+    is_on_gpu = where == "gpu"
+    is_on_cpu = where == "cpu"
+    if is_on_cpu:
+        G.add_edge(10, 20)
+    assert G._is_on_gpu == is_on_gpu
+    assert G._is_on_cpu == is_on_cpu
+    getattr(G, attr)(*args)
+    assert G._is_on_gpu == is_on_gpu
+    assert G._is_on_cpu == is_on_cpu
+    # Also usable from the class and dispatches correctly
+    func = getattr(create_using, attr)
+    func(G, *args)
+    assert G._is_on_gpu == is_on_gpu
+    assert G._is_on_cpu == is_on_cpu
+    # Basic "looks like networkx" checks
+    nx_class = create_using.to_networkx_class()
+    nx_func = getattr(nx_class, attr)
+    assert func.__name__ == nx_func.__name__
+    assert func.__module__.startswith("nx_cugraph")
diff --git a/nx_cugraph/utils/decorators.py b/nx_cugraph/utils/decorators.py
index 15cb010e9..26fce7350 100644
--- a/nx_cugraph/utils/decorators.py
+++ b/nx_cugraph/utils/decorators.py
@@ -167,3 +167,31 @@ def _default_should_run(*args, **kwargs):
 
 def _restore_networkx_dispatched(name):
     return getattr(BackendInterface, name)
+
+
+def _gpu_cpu_api(nx_class, module_name):
+    def _gpu_cpu_graph_method(attr, *, edge_data=False, node_data=False):
+        """Dispatch property to NetworkX or CudaGraph based on cache.
+
+        For example, this will use any cached CudaGraph for ``len(G)``, which
+        prevents creating NetworkX data structures.
+        """
+        nx_func = getattr(nx_class, attr)
+
+        def inner(self, *args, **kwargs):
+            cuda_graph = self._get_cudagraph(edge_data=edge_data, node_data=node_data)
+            if cuda_graph is None:
+                return nx_func(self, *args, **kwargs)
+            return getattr(cuda_graph, attr)(*args, **kwargs)
+
+        inner.__name__ = nx_func.__name__
+        inner.__doc__ = nx_func.__doc__
+        inner.__qualname__ = nx_func.__qualname__
+        inner.__defaults__ = nx_func.__defaults__
+        inner.__kwdefaults__ = nx_func.__kwdefaults__
+        inner.__module__ = module_name
+        inner.__dict__.update(nx_func.__dict__)
+        inner.__wrapped__ = nx_func
+        return inner
+
+    return _gpu_cpu_graph_method