diff --git a/src/taskgraph/graph.py b/src/taskgraph/graph.py
index c0c089035..9951c696f 100644
--- a/src/taskgraph/graph.py
+++ b/src/taskgraph/graph.py
@@ -4,11 +4,17 @@
 
 
 import collections
+import functools
 from dataclasses import dataclass
 
 
 @dataclass(frozen=True)
-class Graph:
+class _Graph:
+    nodes: frozenset
+    edges: frozenset
+
+
+class Graph(_Graph):
     """Generic representation of a directed acyclic graph with labeled edges
     connecting the nodes. Graph operations are implemented in a functional
     manner, so the data structure is immutable.
@@ -23,8 +29,8 @@ class Graph:
     node `left` to node `right`..
     """
 
-    nodes: frozenset
-    edges: frozenset
+    def __init__(self, nodes, edges):
+        super().__init__(frozenset(nodes), frozenset(edges))
 
     def transitive_closure(self, nodes, reverse=False):
         """Return the transitive closure of <nodes>: the graph containing all
@@ -67,23 +73,28 @@ def transitive_closure(self, nodes, reverse=False):
             add_nodes = {(left if reverse else right) for (left, right, _) in add_edges}
             new_nodes = nodes | add_nodes
             new_edges = edges | add_edges
-        return Graph(new_nodes, new_edges)  # type: ignore
+        return Graph(new_nodes, new_edges)
 
     def _visit(self, reverse):
-        queue = collections.deque(sorted(self.nodes))
-        links_by_node = self.reverse_links_dict() if reverse else self.links_dict()
-        seen = set()
+        forward_links, reverse_links = self.links_and_reverse_links_dict()
+
+        dependencies = reverse_links if reverse else forward_links
+        dependents = forward_links if reverse else reverse_links
+
+        indegree = {node: len(dependencies[node]) for node in self.nodes}
+
+        queue = collections.deque(
+            node for node, degree in indegree.items() if degree == 0
+        )
+
         while queue:
             node = queue.popleft()
-            if node in seen:
-                continue
-            links = links_by_node[node]
-            if all((n in seen) for n in links):
-                seen.add(node)
-                yield node
-            else:
-                queue.extend(n for n in links if n not in seen)
-                queue.append(node)
+            yield node
+
+            for dependent in dependents[node]:
+                indegree[dependent] -= 1
+                if indegree[dependent] == 0:
+                    queue.append(dependent)
 
     def visit_postorder(self):
         """
@@ -102,6 +113,21 @@ def visit_preorder(self):
         """
         return self._visit(True)
 
+    @functools.cache
+    def links_and_reverse_links_dict(self):
+        """
+        Return both links and reverse_links dictionaries.
+        Returns a (forward_links, reverse_links) tuple where forward_links maps
+        each node to the set of nodes it links to, and reverse_links maps each
+        node to the set of nodes linking to it.
+        """
+        forward = collections.defaultdict(set)
+        reverse = collections.defaultdict(set)
+        for left, right, _ in self.edges:
+            forward[left].add(right)
+            reverse[right].add(left)
+        return (forward, reverse)
+
     def links_dict(self):
         """
         Return a dictionary mapping each node to a set of the nodes it links to
diff --git a/src/taskgraph/morph.py b/src/taskgraph/morph.py
index 01af7778f..357741f24 100644
--- a/src/taskgraph/morph.py
+++ b/src/taskgraph/morph.py
@@ -51,7 +51,7 @@ def amend_taskgraph(taskgraph, label_to_taskid, to_add):
         for depname, dep in task.dependencies.items():
             new_edges.add((task.task_id, dep, depname))
 
-    taskgraph = TaskGraph(new_tasks, Graph(set(new_tasks), new_edges))  # type: ignore
+    taskgraph = TaskGraph(new_tasks, Graph(frozenset(new_tasks), new_edges))
     return taskgraph, label_to_taskid
 
 
diff --git a/src/taskgraph/optimize/base.py b/src/taskgraph/optimize/base.py
index 34d54f503..3d2776a87 100644
--- a/src/taskgraph/optimize/base.py
+++ b/src/taskgraph/optimize/base.py
@@ -451,7 +451,9 @@ def get_subgraph(
         if left in tasks_by_taskid and right in tasks_by_taskid
     }
 
-    return TaskGraph(tasks_by_taskid, Graph(set(tasks_by_taskid), edges_by_taskid))  # type: ignore
+    return TaskGraph(
+        tasks_by_taskid, Graph(frozenset(tasks_by_taskid), edges_by_taskid)
+    )
 
 
 @register_strategy("never")
diff --git a/src/taskgraph/taskgraph.py b/src/taskgraph/taskgraph.py
index a94f615a2..839fcb385 100644
--- a/src/taskgraph/taskgraph.py
+++ b/src/taskgraph/taskgraph.py
@@ -67,5 +67,5 @@ def from_json(cls, tasks_dict):
                 tasks[key].task_id = value["task_id"]
             for depname, dep in value["dependencies"].items():
                 edges.add((key, dep, depname))
-        task_graph = cls(tasks, Graph(set(tasks), edges))  # type: ignore
+        task_graph = cls(tasks, Graph(frozenset(tasks), edges))
         return tasks, task_graph