diff --git a/causallearn/search/ConstraintBased/FCI.py b/causallearn/search/ConstraintBased/FCI.py
index 8a936c8c..cae3d4d8 100644
--- a/causallearn/search/ConstraintBased/FCI.py
+++ b/causallearn/search/ConstraintBased/FCI.py
@@ -829,7 +829,8 @@ def _contains_all(set_a: Set[Node], set_b: Set[Node]):
 
 
 def fci(dataset: ndarray, independence_test_method: str=fisherz, alpha: float = 0.05, depth: int = -1,
-        max_path_length: int = -1, verbose: bool = False, background_knowledge: BackgroundKnowledge | None = None, show_progress: bool = True,
+        max_path_length: int = -1, verbose: bool = False, background_knowledge: BackgroundKnowledge | None = None, 
+        show_progress: bool = True, node_names = None,
         **kwargs) -> Tuple[Graph, List[Edge]]:
     """
     Perform Fast Causal Inference (FCI) algorithm for causal discovery
@@ -884,8 +885,10 @@ def fci(dataset: ndarray, independence_test_method: str=fisherz, alpha: float =
 
 
     nodes = []
+    if node_names is None:
+        node_names = [f"X{i + 1}" for i in range(dataset.shape[1])]
     for i in range(dataset.shape[1]):
-        node = GraphNode(f"X{i + 1}")
+        node = GraphNode(node_names[i])
         node.add_attribute("id", i)
         nodes.append(node)
 
diff --git a/causallearn/utils/PCUtils/BackgroundKnowledge.py b/causallearn/utils/PCUtils/BackgroundKnowledge.py
index 4c6bca0d..19248d48 100644
--- a/causallearn/utils/PCUtils/BackgroundKnowledge.py
+++ b/causallearn/utils/PCUtils/BackgroundKnowledge.py
@@ -165,7 +165,7 @@ def is_forbidden(self, node1: Node, node2: Node) -> bool:
 
         # then check in tier_map
         if self.tier_value_map.keys().__contains__(node1) and self.tier_value_map.keys().__contains__(node2):
-            if self.tier_value_map.get(node1) >= self.tier_value_map.get(node2):
+            if self.tier_value_map.get(node1) > self.tier_value_map.get(node2): # Allow orientation within the same tier
                 return True
 
         return False