diff --git a/dwave_networkx/algorithms/tsp.py b/dwave_networkx/algorithms/tsp.py
index 32fc46de..c453c3e1 100644
--- a/dwave_networkx/algorithms/tsp.py
+++ b/dwave_networkx/algorithms/tsp.py
@@ -114,7 +114,7 @@ def traveling_salesperson(G, sampler=None, lagrange=None, weight='weight',
 traveling_salesman = traveling_salesperson
 
 
-def traveling_salesperson_qubo(G, lagrange=None, weight='weight', missing_edge_penalty='None'):
+def traveling_salesperson_qubo(G, lagrange=None, weight='weight', missing_edge_weight=None):
     """Return the QUBO with ground states corresponding to a minimum TSP route.
 
     If :math:`|G|` is the number of nodes in the graph, the resulting qubo will have:
@@ -134,8 +134,10 @@ def traveling_salesperson_qubo(G, lagrange=None, weight='weight', missing_edge_p
     weight : optional (default 'weight')
         The name of the edge attribute containing the weight.
     
-    missing_edge_penalty : number, optional (default 'sum')
-        For bi-directional graphs, the penalty associated with the back missing edges. Default is none. Alternatives include using the sum all weights.
+    missing_edge_weight : number, optional (default None)
+        For bi-directional graphs, the weight given to missing edges.
+        If None is given (the default), missing edges will be set to
+        the sum of all weights.
 
     Returns
     -------
@@ -158,15 +160,14 @@ def traveling_salesperson_qubo(G, lagrange=None, weight='weight', missing_edge_p
         else:
             lagrange = 2
     
-    missing_edge_weight=""
-    
-    # default penalty format is sum
-    if missing_edge_penalty == "sum":
-        missing_edge_weight = sum(weight for _, _, weight in G.edges.data('weight', default=0))
+    # calculate default missing_edge_weight if required
+    if missing_edge_weight is None:
+        # networkx method to calculate sum of all weights
+        missing_edge_weight = G.size(weight=weight)
 
     # some input checking
     if N in (1, 2):
-        msg = "graph must be a complete graph"
+        msg = "graph must have at least 3 nodes or be empty"
         raise ValueError(msg)
 
     # Creating the QUBO
@@ -194,10 +195,10 @@ def traveling_salesperson_qubo(G, lagrange=None, weight='weight', missing_edge_p
             nextpos = (pos + 1) % N
 
             # going from u -> v
-            Q[((u, pos), (v, nextpos))] += G[u][v].get('weight', missing_edge_weight)
+            Q[((u, pos), (v, nextpos))] += G[u][v].get(weight, missing_edge_weight)
 
             # going from v -> u
-            Q[((v, pos), (u, nextpos))] += G[u][v].get('weight', missing_edge_weight)
+            Q[((v, pos), (u, nextpos))] += G[u][v].get(weight, missing_edge_weight)
 
     return Q
 
diff --git a/tests/test_tsp.py b/tests/test_tsp.py
index 048acedd..c0bd6532 100644
--- a/tests/test_tsp.py
+++ b/tests/test_tsp.py
@@ -97,6 +97,29 @@ def test_start(self):
 
         self.assertEqual(route[0], 1)
 
+    def test_weighted_complete_digraph(self):
+        G = nx.DiGraph()
+        G.add_weighted_edges_from([
+            (0, 1, 2),
+            (1, 0, 1),
+            (0, 2, 2),
+            (2, 0, 2),
+            (0, 3, 1),
+            (3, 0, 2),
+            (1, 2, 2),
+            (2, 1, 1),
+            (1, 3, 2),
+            (3, 1, 2),
+            (2, 3, 2),
+            (3, 2, 1),
+        ])
+
+        route = dnx.traveling_salesperson(G, dimod.ExactSolver(), start=1)
+
+        self.assertEqual(len(route), len(G))
+        self.assertEqual(nx.path_weight(G, route, 'weight'), 4)
+        self.assertListEqual(route, [1, 0, 3, 2])
+
 
 class TestTSPQUBO(unittest.TestCase):
     def test_empty(self):
@@ -137,7 +160,7 @@ def test_k3(self):
         self.assertEqual(ground_count, len(min_routes))
 
     def test_k3_bidirectional(self):
-        G = nx.Graph()
+        G = nx.DiGraph()
         G.add_weighted_edges_from([('a', 'b', 0.5),
                                    ('b', 'c', 1.0),
                                    ('a', 'c', 2.0),
@@ -171,6 +194,26 @@ def test_k3_bidirectional(self):
 
         self.assertEqual(ground_count, len(min_routes))
 
+    def test_graph_missing_edges(self):
+        G = nx.Graph()
+        G.add_weighted_edges_from([
+            (1, 0, 1),
+            (0, 3, 1),
+            (3, 2, 1),
+            (2, 1, 1),
+        ])
+        dnx.traveling_salesperson_qubo(G, missing_edge_weight=15)
+
+    def test_digraph_missing_edges(self):
+        G = nx.DiGraph()
+        G.add_weighted_edges_from([
+            (1, 0, 1),
+            (0, 3, 1),
+            (3, 2, 1),
+            (2, 1, 1),
+        ])
+        dnx.traveling_salesperson_qubo(G, missing_edge_weight=15)
+
     def test_k4_equal_weights(self):
         # k5 with all equal weights so all paths are equally good
         G = nx.Graph()