Update _arrow_string_mixins.py

Address pandas-dev#61072.

Author: ptth222

pandas/core/arrays/_arrow_string_mixins.py

@@ -321,8 +321,12 @@ def _str_fullmatch(
         flags: int = 0,
         na: Scalar | lib.NoDefault = lib.no_default,
     ):
-        if not pat.endswith("$") or pat.endswith("\\$"):
-            pat = f"{pat}$"
+        if not pat.endswith("$") and not pat.startswith("^"):
+            pat = f"^({pat})$"
+        elif not pat.endswith("$"):
+            pat = f"^({pat[1:]})$"
+        elif not pat.startswith("^"):
+            pat = f"^({pat[0:-1]})$"
         return self._str_match(pat, case, flags, na)
 
     def _str_find(self, sub: str, start: int = 0, end: int | None = None):