diff --git a/src/sage/combinat/rigged_configurations/bij_abstract_class.py b/src/sage/combinat/rigged_configurations/bij_abstract_class.py
index 10140d1b6b2..30292da22ee 100644
--- a/src/sage/combinat/rigged_configurations/bij_abstract_class.py
+++ b/src/sage/combinat/rigged_configurations/bij_abstract_class.py
@@ -311,6 +311,11 @@ def __init__(self, RC_element):
         # TODO: Convert from cur_partitions to rigged_con
         self.cur_partitions = deepcopy(list(self.rigged_con)[:])
 
+        # This is a dummy edge to start the process
+        cp = RC_element.__copy__()
+        cp.set_immutable()
+        self._graph = [ [[], (cp, 0)] ]
+
         # Compute the current L matrix
 #        self.L = {}
 #        for dim in self.rigged_con.parent().dims:
@@ -341,15 +346,17 @@ def __eq__(self, rhs):
         """
         return isinstance(rhs, RCToKRTBijectionAbstract)
 
-    def run(self, verbose=False):
+    def run(self, verbose=False, build_graph=False):
         """
         Run the bijection from rigged configurations to tensor product of KR
         tableaux.
 
         INPUT:
 
-        - ``verbose`` -- (Default: ``False``) Display each step in the
+        - ``verbose`` -- (default: ``False``) display each step in the
           bijection
+        - ``build_graph`` -- (default: ``False``) build the graph of each
+          step in the bijection
 
         EXAMPLES::
 
@@ -390,6 +397,10 @@ def run(self, verbose=False):
                     for a in range(self.n):
                         self._update_vacancy_numbers(a)
 
+                    if build_graph:
+                        y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                        self._graph.append([self._graph[-1][1], (y, len(self._graph)), 'ls'])
+
                 while self.cur_dims[0][0] > 0:
                     if verbose:
                         print("====================")
@@ -404,8 +415,20 @@ def run(self, verbose=False):
                     # Make sure we have a crystal letter
                     ret_crystal_path[-1].append(letters(b)) # Append the rank
 
+                    if build_graph:
+                        y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                        self._graph.append([self._graph[-1][1], (y, len(self._graph)), letters(b)])
+
                 self.cur_dims.pop(0) # Pop off the leading column
 
+        if build_graph:
+            self._graph.pop(0) # Remove the dummy at the start
+            from sage.graphs.digraph import DiGraph
+            from sage.graphs.dot2tex_utils import have_dot2tex
+            self._graph = DiGraph(self._graph)
+            if have_dot2tex():
+                self._graph.set_latex_options(format="dot2tex", edge_labels=True)
+
         # Basic check to make sure we end with the empty configuration
         #tot_len = sum([len(rp) for rp in self.cur_partitions])
         #if tot_len != 0:
diff --git a/src/sage/combinat/rigged_configurations/bij_type_B.py b/src/sage/combinat/rigged_configurations/bij_type_B.py
index a1c6fa028cb..ea1e3ea0192 100644
--- a/src/sage/combinat/rigged_configurations/bij_type_B.py
+++ b/src/sage/combinat/rigged_configurations/bij_type_B.py
@@ -524,15 +524,17 @@ class RCToKRTBijectionTypeB(RCToKRTBijectionTypeC):
     Specific implementation of the bijection from rigged configurations to
     tensor products of KR tableaux for type `B_n^{(1)}`.
     """
-    def run(self, verbose=False):
+    def run(self, verbose=False, build_graph=False):
         """
         Run the bijection from rigged configurations to tensor product of KR
         tableaux for type `B_n^{(1)}`.
 
         INPUT:
 
-        - ``verbose`` -- (Default: ``False``) Display each step in the
+        - ``verbose`` -- (default: ``False``) display each step in the
           bijection
+        - ``build_graph`` -- (default: ``False``) build the graph of each
+          step in the bijection
 
         EXAMPLES::
 
@@ -589,6 +591,10 @@ def run(self, verbose=False):
                         bij.cur_partitions[i]._list[j] *= 2
                         bij.cur_partitions[i].rigging[j] *= 2
                         bij.cur_partitions[i].vacancy_numbers[j] *= 2
+
+                if build_graph:
+                    y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                    self._graph.append([self._graph[-1][1], (y, len(self._graph)), '2x'])
         
                 # Perform the type A_{2n-1}^{(2)} bijection
         
@@ -603,6 +609,10 @@ def run(self, verbose=False):
                         # All it does is update the vacancy numbers on the RC side
                         for a in range(self.n):
                             bij._update_vacancy_numbers(a)
+
+                        if build_graph:
+                            y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                            self._graph.append([self._graph[-1][1], (y, len(self._graph)), 'ls'])
         
                     while bij.cur_dims[0][0] > 0:
                         if verbose:
@@ -617,6 +627,10 @@ def run(self, verbose=False):
                         # Make sure we have a crystal letter
                         ret_crystal_path[-1].append(letters(b)) # Append the rank
 
+                        if build_graph:
+                            y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                            self._graph.append([self._graph[-1][1], (y, len(self._graph)), letters(b)])
+
                     bij.cur_dims.pop(0) # Pop off the leading column
 
                 self.cur_dims.pop(0) # Pop off the spin rectangle
@@ -639,6 +653,10 @@ def run(self, verbose=False):
                         self.cur_partitions[i]._list[j] //= 2
                         self.cur_partitions[i].rigging[j] //= 2
                         self.cur_partitions[i].vacancy_numbers[j] //= 2
+
+                if build_graph:
+                    y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                    self._graph.append([self._graph[-1][1], (y, len(self._graph)), '1/2x'])
             else:
                 # Perform the regular type B_n^{(1)} bijection
 
@@ -662,6 +680,10 @@ def run(self, verbose=False):
                         for a in range(self.n):
                             self._update_vacancy_numbers(a)
 
+                        if build_graph:
+                            y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                            self._graph.append([self._graph[-1][1], (y, len(self._graph)), '2x'])
+
                     while self.cur_dims[0][0] > 0:
                         if verbose:
                             print("====================")
@@ -676,8 +698,20 @@ def run(self, verbose=False):
                         # Make sure we have a crystal letter
                         ret_crystal_path[-1].append(letters(b)) # Append the rank
 
+                        if build_graph:
+                            y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                            self._graph.append([self._graph[-1][1], (y, len(self._graph)), letters(b)])
+
                     self.cur_dims.pop(0) # Pop off the leading column
 
+        if build_graph:
+            self._graph.pop(0) # Remove the dummy at the start
+            from sage.graphs.digraph import DiGraph
+            from sage.graphs.dot2tex_utils import have_dot2tex
+            self._graph = DiGraph(self._graph)
+            if have_dot2tex():
+                self._graph.set_latex_options(format="dot2tex", edge_labels=True)
+
         return self.KRT(pathlist=ret_crystal_path)
 
     def next_state(self, height):
diff --git a/src/sage/combinat/rigged_configurations/bij_type_D.py b/src/sage/combinat/rigged_configurations/bij_type_D.py
index 4da88876953..f8b1a19cd4f 100644
--- a/src/sage/combinat/rigged_configurations/bij_type_D.py
+++ b/src/sage/combinat/rigged_configurations/bij_type_D.py
@@ -414,15 +414,17 @@ class RCToKRTBijectionTypeD(RCToKRTBijectionTypeA):
     r"""
     Specific implementation of the bijection from rigged configurations to tensor products of KR tableaux for type `D_n^{(1)}`.
     """
-    def run(self, verbose=False):
+    def run(self, verbose=False, build_graph=False):
         """
         Run the bijection from rigged configurations to tensor product of KR
         tableaux for type `D_n^{(1)}`.
 
         INPUT:
 
-        - ``verbose`` -- (Default: ``False``) Display each step in the
+        - ``verbose`` -- (default: ``False``) display each step in the
           bijection
+        - ``build_graph`` -- (default: ``False``) build the graph of each
+          step in the bijection
 
         EXAMPLES::
 
@@ -455,6 +457,10 @@ def run(self, verbose=False):
                     for a in range(self.n):
                         self._update_vacancy_numbers(a)
 
+                    if build_graph:
+                        y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                        self._graph.append([self._graph[-1][1], (y, len(self._graph)), 'ls'])
+
                 # Check to see if we are a spinor
                 if dim[0] >= self.n - 1:
                     if verbose:
@@ -465,6 +471,11 @@ def run(self, verbose=False):
                         print("--------------------\n")
                         print("Applying doubling map")
                     self.doubling_map()
+
+                    if build_graph:
+                        y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                        self._graph.append([self._graph[-1][1], (y, len(self._graph)), '2x'])
+
                     if dim[0] == self.n - 1:
                         if verbose:
                             print("====================")
@@ -477,6 +488,10 @@ def run(self, verbose=False):
                             b = -self.n
                         ret_crystal_path[-1].append(letters(b)) # Append the rank
 
+                        if build_graph:
+                            y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                            self._graph.append([self._graph[-1][1], (y, len(self._graph)), letters(b)])
+
                 while self.cur_dims[0][0] > 0:
                     if verbose:
                         print("====================")
@@ -496,6 +511,10 @@ def run(self, verbose=False):
                     # Make sure we have a crystal letter
                     ret_crystal_path[-1].append(letters(b)) # Append the rank
 
+                    if build_graph:
+                        y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                        self._graph.append([self._graph[-1][1], (y, len(self._graph)), letters(b)])
+
                 self.cur_dims.pop(0) # Pop off the leading column
 
                 # Check to see if we were a spinor
@@ -508,6 +527,19 @@ def run(self, verbose=False):
                         print("--------------------\n")
                         print("Applying halving map")
                     self.halving_map()
+
+                    if build_graph:
+                        y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                        self._graph.append([self._graph[-1][1], (y, len(self._graph)), '1/2x'])
+
+        if build_graph:
+            self._graph.pop(0) # Remove the dummy at the start
+            from sage.graphs.digraph import DiGraph
+            from sage.graphs.dot2tex_utils import have_dot2tex
+            self._graph = DiGraph(self._graph)
+            if have_dot2tex():
+                self._graph.set_latex_options(format="dot2tex", edge_labels=True)
+
         return self.KRT(pathlist=ret_crystal_path)
 
     def next_state(self, height):
diff --git a/src/sage/combinat/rigged_configurations/bij_type_D_twisted.py b/src/sage/combinat/rigged_configurations/bij_type_D_twisted.py
index 0796a8f6819..accf74e60bc 100644
--- a/src/sage/combinat/rigged_configurations/bij_type_D_twisted.py
+++ b/src/sage/combinat/rigged_configurations/bij_type_D_twisted.py
@@ -310,15 +310,17 @@ class RCToKRTBijectionTypeDTwisted(RCToKRTBijectionTypeD, RCToKRTBijectionTypeA2
     Specific implementation of the bijection from rigged configurations to
     tensor products of KR tableaux for type `D_{n+1}^{(2)}`.
     """
-    def run(self, verbose=False):
+    def run(self, verbose=False, build_graph=False):
         """
         Run the bijection from rigged configurations to tensor product of KR
         tableaux for type `D_{n+1}^{(2)}`.
 
         INPUT:
 
-        - ``verbose`` -- (Default: ``False``) Display each step in the
+        - ``verbose`` -- (default: ``False``) display each step in the
           bijection
+        - ``build_graph`` -- (default: ``False``) build the graph of each
+          step in the bijection
 
         EXAMPLES::
 
@@ -351,6 +353,10 @@ def run(self, verbose=False):
                     for a in range(self.n):
                         self._update_vacancy_numbers(a)
 
+                    if build_graph:
+                        y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                        self._graph.append([self._graph[-1][1], (y, len(self._graph)), 'ls'])
+
                 # Check to see if we are a spinor
                 if dim[0] == self.n:
                     if verbose:
@@ -362,6 +368,10 @@ def run(self, verbose=False):
                         print("Applying doubling map")
                     self.doubling_map()
 
+                    if build_graph:
+                        y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                        self._graph.append([self._graph[-1][1], (y, len(self._graph)), '2x'])
+
                 while self.cur_dims[0][0] > 0:
                     if verbose:
                         print("====================")
@@ -376,6 +386,10 @@ def run(self, verbose=False):
                     # Make sure we have a crystal letter
                     ret_crystal_path[-1].append(letters(b)) # Append the rank
 
+                    if build_graph:
+                        y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                        self._graph.append([self._graph[-1][1], (y, len(self._graph)), letters(b)])
+
                 self.cur_dims.pop(0) # Pop off the leading column
 
                 # Check to see if we were a spinor
@@ -388,6 +402,19 @@ def run(self, verbose=False):
                         print("--------------------\n")
                         print("Applying halving map")
                     self.halving_map()
+
+                    if build_graph:
+                        y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions])
+                        self._graph.append([self._graph[-1][1], (y, len(self._graph)), '1/2x'])
+
+        if build_graph:
+            self._graph.pop(0) # Remove the dummy at the start
+            from sage.graphs.digraph import DiGraph
+            from sage.graphs.dot2tex_utils import have_dot2tex
+            self._graph = DiGraph(self._graph)
+            if have_dot2tex():
+                self._graph.set_latex_options(format="dot2tex", edge_labels=True)
+
         return self.KRT(pathlist=ret_crystal_path)
 
     def next_state(self, height):
diff --git a/src/sage/combinat/rigged_configurations/rigged_configuration_element.py b/src/sage/combinat/rigged_configurations/rigged_configuration_element.py
index 5c845cf611b..9e10c024da2 100644
--- a/src/sage/combinat/rigged_configurations/rigged_configuration_element.py
+++ b/src/sage/combinat/rigged_configurations/rigged_configuration_element.py
@@ -493,7 +493,7 @@ def check(self):
                 if vac_num < partition.rigging[i]:
                     raise ValueError("rigging can be at most the vacancy number")
 
-    def to_tensor_product_of_kirillov_reshetikhin_tableaux(self, display_steps=False):
+    def to_tensor_product_of_kirillov_reshetikhin_tableaux(self, display_steps=False, build_graph=False):
         r"""
         Perform the bijection from this rigged configuration to a tensor
         product of Kirillov-Reshetikhin tableaux given in [RigConBijection]_
@@ -510,6 +510,8 @@ def to_tensor_product_of_kirillov_reshetikhin_tableaux(self, display_steps=False
 
         - ``display_steps`` -- (default: ``False``) boolean which indicates
           if we want to output each step in the algorithm
+        - ``build_graph` -- (default: ``False``) boolean which indicates
+          if we want to construct and return a graph of the bijection
 
         OUTPUT:
 
@@ -544,7 +546,11 @@ def to_tensor_product_of_kirillov_reshetikhin_tableaux(self, display_steps=False
             True
         """
         from sage.combinat.rigged_configurations.bijection import RCToKRTBijection
-        return RCToKRTBijection(self).run(display_steps)
+        bij = RCToKRTBijection(self)
+        ret = bij.run(display_steps, build_graph)
+        if build_graph:
+            return (ret, bij._graph)
+        return ret
 
     def to_tensor_product_of_kirillov_reshetikhin_crystals(self, display_steps=False):
         r"""