diff --git a/pytensor/graph/basic.py b/pytensor/graph/basic.py
index f173b2fe96..65fe3750b9 100644
--- a/pytensor/graph/basic.py
+++ b/pytensor/graph/basic.py
@@ -22,6 +22,7 @@
     TypeVar,
     Union,
     cast,
+    overload,
 )
 
 import numpy as np
@@ -1301,9 +1302,31 @@ def clone_get_equiv(
     return memo
 
 
+@overload
+def general_toposort(
+    outputs: Iterable[T],
+    deps: None,
+    compute_deps_cache: Callable[[T], Optional[Union[OrderedSet, list[T]]]],
+    deps_cache: Optional[dict[T, list[T]]],
+    clients: Optional[dict[T, list[T]]],
+) -> list[T]:
+    ...
+
+
+@overload
 def general_toposort(
     outputs: Iterable[T],
     deps: Callable[[T], Union[OrderedSet, list[T]]],
+    compute_deps_cache: None,
+    deps_cache: None,
+    clients: Optional[dict[T, list[T]]],
+) -> list[T]:
+    ...
+
+
+def general_toposort(
+    outputs: Iterable[T],
+    deps: Optional[Callable[[T], Union[OrderedSet, list[T]]]],
     compute_deps_cache: Optional[
         Callable[[T], Optional[Union[OrderedSet, list[T]]]]
     ] = None,
@@ -1345,7 +1368,7 @@ def general_toposort(
         if deps_cache is None:
             deps_cache = {}
 
-        def _compute_deps_cache(io):
+        def _compute_deps_cache_(io):
             if io not in deps_cache:
                 d = deps(io)
 
@@ -1363,6 +1386,8 @@ def _compute_deps_cache(io):
             else:
                 return deps_cache[io]
 
+        _compute_deps_cache = _compute_deps_cache_
+
     else:
         _compute_deps_cache = compute_deps_cache
 
@@ -1451,15 +1476,14 @@ def io_toposort(
                 )
         return order
 
-    compute_deps = None
-    compute_deps_cache = None
     iset = set(inputs)
-    deps_cache: dict = {}
 
     if not orderings:  # ordering can be None or empty dict
         # Specialized function that is faster when no ordering.
         # Also include the cache in the function itself for speed up.
 
+        deps_cache: dict = {}
+
         def compute_deps_cache(obj):
             if obj in deps_cache:
                 return deps_cache[obj]
@@ -1478,6 +1502,14 @@ def compute_deps_cache(obj):
                 deps_cache[obj] = rval
             return rval
 
+        topo = general_toposort(
+            outputs,
+            deps=None,
+            compute_deps_cache=compute_deps_cache,
+            deps_cache=deps_cache,
+            clients=clients,
+        )
+
     else:
         # the inputs are used only here in the function that decides what
         # 'predecessors' to explore
@@ -1494,13 +1526,13 @@ def compute_deps(obj):
                 assert not orderings.get(obj, None)
             return rval
 
-    topo = general_toposort(
-        outputs,
-        deps=compute_deps,
-        compute_deps_cache=compute_deps_cache,
-        deps_cache=deps_cache,
-        clients=clients,
-    )
+        topo = general_toposort(
+            outputs,
+            deps=compute_deps,
+            compute_deps_cache=None,
+            deps_cache=None,
+            clients=clients,
+        )
     return [o for o in topo if isinstance(o, Apply)]
 
 
diff --git a/pytensor/graph/rewriting/basic.py b/pytensor/graph/rewriting/basic.py
index c24cd99358..d334b9894d 100644
--- a/pytensor/graph/rewriting/basic.py
+++ b/pytensor/graph/rewriting/basic.py
@@ -2405,13 +2405,15 @@ def importer(node):
                 if node is not current_node:
                     q.append(node)
 
-            chin = None
+            chin: Optional[Callable] = None
             if self.tracks_on_change_inputs:
 
-                def chin(node, i, r, new_r, reason):
+                def chin_(node, i, r, new_r, reason):
                     if node is not current_node and not isinstance(node, str):
                         q.append(node)
 
+                chin = chin_
+
             u = self.attach_updater(
                 fgraph, importer, None, chin=chin, name=getattr(self, "name", None)
             )
diff --git a/tests/tensor/test_math.py b/tests/tensor/test_math.py
index 045a776081..337ef20685 100644
--- a/tests/tensor/test_math.py
+++ b/tests/tensor/test_math.py
@@ -1403,7 +1403,7 @@ def test_bool(self):
 
 
 rng = np.random.default_rng(seed=utt.fetch_seed())
-TestClip = makeTester(
+TestClip1 = makeTester(
     name="ClipTester",
     op=clip,
     expected=lambda x, y, z: np.clip(x, y, z),
@@ -1470,7 +1470,7 @@ def test_bool(self):
 )
 
 
-class TestClip:
+class TestClip2:
     def test_complex_value(self):
         for dtype in ["complex64", "complex128"]:
             a = vector(dtype=dtype)