diff --git a/fastapi/_compat/v2.py b/fastapi/_compat/v2.py
index 6a87b9ae9..af161261d 100644
--- a/fastapi/_compat/v2.py
+++ b/fastapi/_compat/v2.py
@@ -262,7 +262,18 @@ def _replace_refs(
     new_schema = deepcopy(schema)
     for key, value in new_schema.items():
         if key == "$ref":
-            ref_name = schema["$ref"].split("/")[-1]
+            #ref_name = schema["$ref"].split("/")[-1]
+            ref_name = schema["$ref"]
+            if isinstance(ref_value, str):
+                # Normal case
+                ref_name = ref_value.split("/")[-1]
+            elif isinstance(ref_value, dict):
+                # Handle dict case 
+                ref_name = ref_value.get("title") or "unknown_ref"
+            else:
+                # Unexpected type
+                ref_name = "invalid_ref"
+            
             if ref_name in old_name_to_new_name_map:
                 new_name = old_name_to_new_name_map[ref_name]
                 new_schema["$ref"] = REF_TEMPLATE.format(model=new_name)