diff --git a/src/taskgraph/graph.py b/src/taskgraph/graph.py
index 576325514..f77d71ef3 100644
--- a/src/taskgraph/graph.py
+++ b/src/taskgraph/graph.py
@@ -107,6 +107,15 @@ def _visit(self, reverse):
                 f"Dependency loop detected involving the following nodes: {loopy_nodes}"
             )
 
+    @functools.cache
+    def _visit_order(self, reverse):
+        """
+        Return the order in which `_visit` yields nodes as a tuple. The graph
+        is immutable, so this is cached to avoid sorting it again every time
+        it is visited.
+        """
+        return tuple(self._visit(reverse))
+
     def visit_postorder(self):
         """
         Generate a sequence of nodes in postorder, such that every node is
@@ -114,14 +123,14 @@ def visit_postorder(self):
 
         Raises an exception if the graph contains a cycle.
         """
-        return self._visit(False)
+        return iter(self._visit_order(False))
 
     def visit_preorder(self):
         """
         Like visit_postorder, but in reverse: evrey node is visited *before*
         any nodes it links to.
         """
-        return self._visit(True)
+        return iter(self._visit_order(True))
 
     @functools.cache
     def links_and_reverse_links_dict(self):
diff --git a/src/taskgraph/optimize/base.py b/src/taskgraph/optimize/base.py
index ba07ca276..aeb6cf5ff 100644
--- a/src/taskgraph/optimize/base.py
+++ b/src/taskgraph/optimize/base.py
@@ -78,7 +78,7 @@ def optimize_task_graph(
 
     # Gather each relevant task's index
     indexes = set()
-    for label in target_task_graph.graph.visit_postorder():
+    for label in target_task_graph.tasks:
         if label in do_not_optimize:
             continue
         _, strategy, arg = optimizations(label)
@@ -157,7 +157,7 @@ def remove_tasks(
     opt_counts = defaultdict(int)
     opt_reasons = {}
     removed = set()
-    dependents_of = target_task_graph.graph.reverse_links_dict()
+    _, dependents_of = target_task_graph.graph.links_and_reverse_links_dict()
     tasks = target_task_graph.tasks
     prune_candidates = set()
 
diff --git a/test/test_graph_perf.py b/test/test_graph_perf.py
index 83d400f1b..6212033fc 100644
--- a/test/test_graph_perf.py
+++ b/test/test_graph_perf.py
@@ -124,6 +124,8 @@ def test_transitive_closure(geometry):
 @pytest.mark.parametrize("geometry", ["linear", "fan", "btree", "diamond"])
 def test_visit_postorder(geometry):
     _, graph, _ = GEOMETRIES[geometry]
+    # Clear the functools.cache to measure actual computation each time
+    graph._visit_order.cache_clear()
     order = list(graph.visit_postorder())
     assert len(order) == N
 
@@ -132,6 +134,8 @@ def test_visit_postorder(geometry):
 @pytest.mark.parametrize("geometry", ["linear", "fan", "btree", "diamond"])
 def test_visit_preorder(geometry):
     _, graph, _ = GEOMETRIES[geometry]
+    # Clear the functools.cache to measure actual computation each time
+    graph._visit_order.cache_clear()
     order = list(graph.visit_preorder())
     assert len(order) == N
 
@@ -161,11 +165,26 @@ def test_links_dict(geometry):
 @pytest.mark.parametrize("geometry", ["linear", "fan", "btree", "diamond"])
 def test_for_each_task(geometry):
     _, _, tg = GEOMETRIES[geometry]
+    # Clear the functools.cache to measure actual computation each time
+    tg.graph._visit_order.cache_clear()
     visited = []
     tg.for_each_task(lambda task, _tg: visited.append(task.label))
     assert len(visited) == N
 
 
+@pytest.mark.benchmark
+@pytest.mark.parametrize("geometry", ["linear", "fan", "btree", "diamond"])
+def test_for_each_task_repeated(geometry):
+    """Visit the same graph several times, like the verifications do."""
+    _, _, tg = GEOMETRIES[geometry]
+    # Clear the functools.cache to measure actual computation each time
+    tg.graph._visit_order.cache_clear()
+    visited = []
+    for _ in range(10):
+        tg.for_each_task(lambda task, _tg: visited.append(task.label))
+    assert len(visited) == 10 * N
+
+
 # ---------------------------------------------------------------------------
 # Benchmarks – TaskGraph JSON round-trip
 # ---------------------------------------------------------------------------
@@ -175,6 +194,8 @@ def test_for_each_task(geometry):
 @pytest.mark.parametrize("geometry", ["linear", "fan", "btree"])
 def test_taskgraph_to_json(geometry):
     _, _, tg = GEOMETRIES[geometry]
+    # Clear the functools.cache to measure actual computation each time
+    tg.graph._visit_order.cache_clear()
     data = tg.to_json()
     assert len(data) == N