diff --git a/content/algorithms/assortativity/correlation.md b/content/algorithms/assortativity/correlation.md
index c5aec79d..fa7a3867 100644
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diff --git a/content/algorithms/dag/index.md b/content/algorithms/dag/index.md
index 6022e17f..f02281d2 100644
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diff --git a/content/algorithms/euler/euler.md b/content/algorithms/euler/euler.md
index f8ff2ac9..eb1777e7 100644
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diff --git a/content/algorithms/flow/dinitz_alg.md b/content/algorithms/flow/dinitz_alg.md
index 9df18d89..4ae67fc0 100644
--- a/content/algorithms/flow/dinitz_alg.md
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diff --git a/content/algorithms/index.md b/content/algorithms/index.md
index cd233db9..ccdd60c1 100644
--- a/content/algorithms/index.md
+++ b/content/algorithms/index.md
@@ -1,5 +1,24 @@
 
 
+
++++
+
++++
+
++++
+
++++
+
++++
+
++++
+
++++
+
++++
+
++++
+
 # Algorithms
 
 A closer look at some of the algorithms and network analysis techniques
@@ -13,4 +32,5 @@ assortativity/correlation
 dag/index
 flow/dinitz_alg
 euler/euler
+vf2_vs_vf2pp/vf2_vs_vf2pp
 ```
diff --git a/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_A.txt b/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_A.txt
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diff --git a/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_graph_indicator.txt b/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_graph_indicator.txt
new file mode 100755
index 00000000..b5741442
--- /dev/null
+++ b/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_graph_indicator.txt
@@ -0,0 +1,43471 @@
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diff --git a/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_graph_labels.txt b/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_graph_labels.txt
new file mode 100755
index 00000000..06b4e6dd
--- /dev/null
+++ b/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_graph_labels.txt
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diff --git a/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_node_attributes.txt b/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_node_attributes.txt
new file mode 100755
index 00000000..2c8d9979
--- /dev/null
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diff --git a/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_node_labels.txt b/content/algorithms/vf2_vs_vf2pp/datasets/PROTEINS_node_labels.txt
new file mode 100755
index 00000000..4f0beae2
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diff --git a/content/algorithms/vf2_vs_vf2pp/datasets/README.txt b/content/algorithms/vf2_vs_vf2pp/datasets/README.txt
new file mode 100755
index 00000000..7efd04b5
--- /dev/null
+++ b/content/algorithms/vf2_vs_vf2pp/datasets/README.txt
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+README for dataset PROTEINS
+
+
+=== Usage ===
+
+This folder contains the following comma separated text files 
+(replace DS by the name of the dataset):
+
+n = total number of nodes
+m = total number of edges
+N = number of graphs
+
+(1) 	DS_A.txt (m lines) 
+	sparse (block diagonal) adjacency matrix for all graphs,
+	each line corresponds to (row, col) resp. (node_id, node_id)
+
+(2) 	DS_graph_indicator.txt (n lines)
+	column vector of graph identifiers for all nodes of all graphs,
+	the value in the i-th line is the graph_id of the node with node_id i
+
+(3) 	DS_graph_labels.txt (N lines) 
+	class labels for all graphs in the dataset,
+	the value in the i-th line is the class label of the graph with graph_id i
+
+(4) 	DS_node_labels.txt (n lines)
+	column vector of node labels,
+	the value in the i-th line corresponds to the node with node_id i
+
+There are OPTIONAL files if the respective information is available:
+
+(5) 	DS_edge_labels.txt (m lines; same size as DS_A_sparse.txt)
+	labels for the edges in DS_A_sparse.txt 
+
+(6) 	DS_edge_attributes.txt (m lines; same size as DS_A.txt)
+	attributes for the edges in DS_A.txt 
+
+(7) 	DS_node_attributes.txt (n lines) 
+	matrix of node attributes,
+	the comma seperated values in the i-th line is the attribute vector of the node with node_id i
+
+(8) 	DS_graph_attributes.txt (N lines) 
+	regression values for all graphs in the dataset,
+	the value in the i-th line is the attribute of the graph with graph_id i
+
+
+=== Previous Use of the Dataset ===
+
+Feragen, A., Kasenburg, N., Petersen, J., de Bruijne, M., Borgwardt, K.M.: Scalable
+kernels for graphs with continuous attributes. In: C.J.C. Burges, L. Bottou, Z. Ghahra-
+mani, K.Q. Weinberger (eds.) NIPS, pp. 216-224 (2013)
+
+Neumann, M., Garnett R., Bauckhage Ch., Kersting K.: Propagation Kernels: Efficient Graph 
+Kernels from Propagated Information. Under review at MLJ.
+
+
+=== References ===
+
+K. M. Borgwardt, C. S. Ong, S. Schoenauer, S. V. N. Vishwanathan, A. J. Smola, and H. P. 
+Kriegel. Protein function prediction via graph kernels. Bioinformatics, 21(Suppl 1):i47–i56, 
+Jun 2005.
+
+P. D. Dobson and A. J. Doig. Distinguishing enzyme structures from non-enzymes without 
+alignments. J. Mol. Biol., 330(4):771–783, Jul 2003.
+
+
diff --git a/content/algorithms/vf2_vs_vf2pp/ex_PROTEINS.png b/content/algorithms/vf2_vs_vf2pp/ex_PROTEINS.png
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diff --git a/content/algorithms/vf2_vs_vf2pp/proteins_dataset_vf2vsvf2pp.py b/content/algorithms/vf2_vs_vf2pp/proteins_dataset_vf2vsvf2pp.py
new file mode 100644
index 00000000..efec368e
--- /dev/null
+++ b/content/algorithms/vf2_vs_vf2pp/proteins_dataset_vf2vsvf2pp.py
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+import numpy as np
+import networkx as nx
+from spektral.datasets import TUDataset
+import matplotlib.pyplot as plt
+import timeit
+import pandas as pd
+
+
+def tud_to_networkx(ds_name):
+    pre = ds_name + "/"
+    
+    with open("datasets/" +   ds_name + "_graph_indicator.txt", "r") as f:
+        graph_indicator = [int(i) - 1 for i in list(f)]
+    f.closed
+
+    # Nodes.
+    num_graphs = max(graph_indicator)
+    node_indices = []
+    offset = []
+    c = 0
+
+    for i in range(num_graphs + 1):
+        offset.append(c)
+        c_i = graph_indicator.count(i)
+        node_indices.append((c, c + c_i - 1))
+        c += c_i
+
+    graph_db = []
+    for i in node_indices:
+        g = nx.Graph()
+        for j in range(i[1] - i[0]+1):
+            g.add_node(j)
+
+        graph_db.append(g)
+
+    # Edges.
+    with open("datasets/" +  ds_name + "_A.txt", "r") as f:
+        edges = [i.split(',') for i in list(f)]
+    f.closed
+
+    edges = [(int(e[0].strip()) - 1, int(e[1].strip()) - 1) for e in edges]
+    edge_list = []
+    edgeb_list = []
+    for e in edges:
+        g_id = graph_indicator[e[0]]
+        g = graph_db[g_id]
+        off = offset[g_id]
+
+        # Avoid multigraph (for edge_list)
+        if ((e[0] - off, e[1] - off) not in list(g.edges())) and ((e[1] - off, e[0] - off) not in list(g.edges())):
+            g.add_edge(e[0] - off, e[1] - off)
+            edge_list.append((e[0] - off, e[1] - off))
+            edgeb_list.append(True)
+        else:
+            edgeb_list.append(False)
+
+    # Node labels.
+    with open("datasets/" +  ds_name + "_node_labels.txt", "r") as f:
+        node_labels = [str.strip(i) for i in list(f)]
+    f.closed
+
+    node_labels = [i.split(',') for i in node_labels]
+    int_labels = [];
+    for i in range(len(node_labels)):
+        int_labels.append([int(j) for j in node_labels[i]])
+
+    i = 0
+    for g in graph_db:
+        for v in range(g.number_of_nodes()):
+            g.nodes[v]['labels'] = int_labels[i]
+            i += 1
+
+    # Node Attributes.
+    with open("datasets/" +  ds_name + "_node_attributes.txt", "r") as f:
+        node_attributes = [str.strip(i) for i in list(f)]
+    f.closed
+
+    node_attributes = [i.split(',') for i in node_attributes]
+    float_attributes = [];
+    for i in range(len(node_attributes)):
+        float_attributes.append([float(j) for j in node_attributes[i]])
+    i = 0
+    for g in graph_db:
+        for v in range(g.number_of_nodes()):
+            g.nodes[v]['attributes'] = float_attributes[i]
+            i += 1
+
+    # Classes.
+    with open("datasets/" +  ds_name + "_graph_labels.txt", "r") as f:
+        classes = [str.strip(i) for i in list(f)]
+    f.closed
+    classes = [i.split(',') for i in classes]
+    cs = [];
+    for i in range(len(classes)):
+        cs.append([int(j) for j in classes[i]])
+
+    i = 0
+    for g in graph_db:
+        g.graph['classes'] = cs[i]
+        i += 1
+
+    return graph_db
+
+def main():
+    n = int(input("How many graphs do you want to test?"))
+    reps = int(input("Enter the number of repetitions"))
+
+    dataset = 'PROTEINS'
+    data = TUDataset(dataset) #Load Dataset 
+    graph_db = tud_to_networkx(dataset) #NetworkX Data
+    
+    #Sort graphs by size 
+    graph_sizes = [len(graph_db[i].nodes) for i in range(0, len(graph_db))]
+    sizes_sorted = np.sort(graph_sizes)
+    
+    order = np.argsort(graph_sizes)
+    graph_db_sorted = np.array(graph_db, dtype=object)[order]
+
+    #Measure time execution of VF2 and VF2++
+    vf2_all_times_graphs = []  # VF2 time measurements on graphs
+    vf2pp_all_times_graphs = []  # VF2++ time measurements on graphs
+
+    for i in range(0,n):
+        G = graph_db_sorted[i]
+
+        med = np.array(np.zeros(reps))
+        for j in range(0, reps):
+            start = timeit.default_timer()
+
+            nx.is_isomorphic(G, G) #VF2
+
+            stop = timeit.default_timer()
+            total_time = stop - start
+            mins, secs = divmod(total_time, 60)
+            med[j] = secs 
+
+        vf2_all_times_graphs.append(np.median(med))
+
+        med = np.array(np.zeros(reps))
+        for j in range(0, reps):
+            start = timeit.default_timer()
+
+            nx.vf2pp_is_isomorphic(G, G) #VF2++
+
+            stop = timeit.default_timer()
+            total_time = stop - start
+            mins, secs = divmod(total_time, 60)
+            med[j] = secs 
+        
+        vf2pp_all_times_graphs.append(np.median(med))
+    
+    #Create dataset with time measurements 
+    p = pd.DataFrame()
+    p["size"] = sizes_sorted[0:n]
+    p["vf2"] = vf2_all_times_graphs
+    p["vf2++"] = vf2pp_all_times_graphs
+
+    #Calculate the median between all values by size 
+    median = p.groupby(['size']).median() 
+
+    # Plot experiment results 
+    
+    plt.plot(median["vf2"], label = "vf2")
+    plt.plot(median["vf2++"], label = "vf2++")
+    plt.legend(title= "Algorithms")
+    plt.title("VF2 vs. VF2++ on PROTEINS dataset", fontweight="bold")
+    plt.ylabel("Execution Time (seconds)")
+    plt.xlabel("Number of nodes")
+    plt.savefig('ex_PROTEINS.png')
+    plt.show()
+
+if __name__ == "__main__":
+    main()
diff --git a/content/algorithms/vf2_vs_vf2pp/same_degree_seq.png b/content/algorithms/vf2_vs_vf2pp/same_degree_seq.png
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diff --git a/content/algorithms/vf2_vs_vf2pp/vf2_vs_vf2pp.md b/content/algorithms/vf2_vs_vf2pp/vf2_vs_vf2pp.md
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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.2
+kernelspec:
+  display_name: Python 3 (ipykernel)
+  language: python
+  name: python3
+---
+
+# VF2 vs VF2++: Comparative analysis of performance
+
++++
+
+Currently, in NetworkX we have two isomorphism algorithms implemented: VF2 and VF2++.
+
+VF2 and VF2++ are recursive algorithms that explore all the possible matching functions between the nodes of the graphs in order to find one that satisfies the isomorphism. As exploring all the matching functions has an exponential cost some cutting rules are used to avoid unfruitful branches.
+
+**What's new on VF2++?**
+- Node ordering: Previously the order in which the nodes were explored was determined by the labels. In this new algorithm, a particular node order is defined for every graph in order to explore the "rarest" nodes first. The "rarest" nodes are defined based on the graph structure.   
+- Simple-to-check cutting rules. These new rules are cheaper than the old ones but are just as effective to determine if a branch is unfruitful.
+- A non-recursive implementation that saves both time and space.
+
+Let's see an example of how to use VF2 and VF2++ to test for isomorphism with NetworkX.
+
+```{code-cell}
+import networkx as nx
+
+G = nx.bull_graph()  # Create a bull graph
+
+print("VF2:  ", nx.is_isomorphic(G, G))  # VF2
+
+print("VF2++: ", nx.vf2pp_is_isomorphic(G, G))  # VF2++
+```
+
+## Analysis: General approach
+To compare the performance of both algorithms we will run the algorithms over many different types of graphs and digraphs. The analysis will be divided as follows:
+
+- Complete Graphs: Graphs and DiGraphs
+- Regular Graphs
+- Random Erdos Renyi Graphs and DiGraphs
+- Random Graphs and DiGraphs with the same degree sequence
+- Proteins Dataset
+
++++
+
+To measure execution time we can use the function *timeit(setup, stmt, number )*. Where *stmt* is the code that we want to time, *setup* is the code needed to run stmt but that we don't want to time and *number* is how many times stmt is run.
+
+Let's see an example:
+
+```{code-cell}
+import timeit
+
+sleep_time = timeit.timeit(
+    setup="import time",  # Setup code
+    stmt="time.sleep(2)",  # Code to be timed
+    number=1,
+)  # Number of times my code is going to be executed
+
+print("I was asleep for " + str(sleep_time) + " seconds")
+```
+
+Given that the state of the CPU can change the execution time of our code it's important to repeat our measurements many times and the consider some way to summarize our data. We can use the median. To achive this we will use *repeat(setup, stmt, timer, number, repeat)*.
+Let's measure how much time it takes to create a balanced tree with networkX. We will repeat it 100 times and then take the median of all measumerements.
+
+```{code-cell}
+import numpy as np
+import time
+
+np.median(
+    timeit.repeat(
+        setup="import networkx as nx",  # Setup not included into
+        stmt="G = nx.balanced_tree(10, 2)",  # Code to be measured
+        timer=time.process_time,  # Only measure time used by the process running this code
+        number=1,
+        repeat=100,
+    )
+)  # repetitions
+```
+
+For more information about timeit: <https://docs.python.org/3/library/timeit.html>
+
++++
+
+**Disclaimer:** The graphs used in the following examples are going to be small because the execution time grows exponentially as we increase the size of the graphs otherwise the examples are going to take too much time to run. In order to do a more exhaustive benchmarking we need to repeat all the measurements many times and use bigger graphs. This notebook does not intend to be a proper benchmarking but rather a general analysis of the performance of both algorithms.
+
+```{code-cell}
+import matplotlib.pyplot as plt
+import matplotlib.ticker as mticker
+```
+
+## Complete Graphs and DiGraphs without labels
+We can compare both algorithms on complete graphs as a way to see how these algorithms behave on dense graphs.
+Let's measure how much it takes for both algorithms to run in complete graphs as we increase the number of nodes in the graphs. For these graphs we are interested in identifying positive instances, in other words, we want to test graphs that are isomorphic.
+
+```{code-cell}
+vf2_all_times_graphs = []  # VF2 time measurements on graphs
+vf2pp_all_times_graphs = []  # VF2++ time measurements on graphs
+
+vf2_all_times_digraphs = []  # VF2 time measurements on digraphs
+vf2pp_all_times_digraphs = []  # VF2++ time measurements on digraphs
+
+for i in range(10, 120, 10):
+    # time vf2 on graphs
+    vf2_time_g = np.median(
+        timeit.repeat(
+            setup="import networkx as nx;G = nx.complete_graph(" + str(i) + ")",
+            stmt="nx.is_isomorphic(G,G)",
+            timer=time.process_time,
+            number=1,
+            repeat=100,
+        )
+    )
+    vf2_all_times_graphs.append(vf2_time_g)
+
+    # time vf2++ on graphs
+    vf2pp_time_g = np.median(
+        timeit.repeat(
+            setup="import networkx as nx;G = nx.complete_graph(" + str(i) + ")",
+            stmt="nx.vf2pp_is_isomorphic(G,G)",
+            timer=time.process_time,
+            number=1,
+            repeat=100,
+        )
+    )
+    vf2pp_all_times_graphs.append(vf2pp_time_g)
+
+    # time vf2 on digraphs
+    vf2_time_d = np.median(
+        timeit.repeat(
+            setup="import networkx as nx;G = nx.complete_graph("
+            + str(i)
+            + ", nx.DiGraph())",
+            stmt="nx.is_isomorphic(G,G)",
+            timer=time.process_time,
+            number=1,
+            repeat=100,
+        )
+    )
+    vf2_all_times_digraphs.append(vf2_time_d)
+
+    # time vf2++ on digraphs
+    vf2pp_time_d = np.median(
+        timeit.repeat(
+            setup="import networkx as nx;G = nx.complete_graph("
+            + str(i)
+            + ", nx.DiGraph())",
+            stmt="nx.vf2pp_is_isomorphic(G,G)",
+            timer=time.process_time,
+            number=1,
+            repeat=100,
+        )
+    )
+    vf2pp_all_times_digraphs.append(vf2pp_time_d)
+
+# Plot results
+fig, ax = plt.subplots(1, 2)
+
+fig.set_size_inches(10, 5)
+
+plt.suptitle("VF2 vs. VF2++ in complete graphs and digraphs", fontweight="bold")
+
+x = list(range(10, 120, 10))  # x labels
+
+# Plot vf2 and vf2++ in graphs
+(vf2_line_g,) = ax[0].plot(
+    x, vf2_all_times_graphs, label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line_g,) = ax[0].plot(
+    x, vf2pp_all_times_graphs, label="vf2++", linestyle="--", marker="o"
+)
+ax[0].set_title("Graphs")
+ax[0].set_ylabel("Execution Time (seconds)")
+ax[0].set_xlabel("Number of nodes")
+
+(vf2_line_d,) = ax[1].plot(
+    x, vf2_all_times_digraphs, label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line_d,) = ax[1].plot(
+    x, vf2pp_all_times_digraphs, label="vf2++", linestyle="--", marker="o"
+)
+ax[1].set_title("DiGraphs")
+ax[1].set_ylabel("Execution Time (seconds)")
+ax[1].set_xlabel("Number of nodes")
+
+ax[0].legend(handles=[vf2_line_g, vf2pp_line_g])
+ax[1].legend(handles=[vf2_line_d, vf2pp_line_d])
+
+fig.tight_layout(pad=2.0)
+
+plt.show()
+```
+
+As expected VF2++ performs better than VF2 as we increase the number of nodes. We can see that the curve of VF2 grows significantly faster compared to the curve of VF2++.  From this experiment, we can't affirm that VF2++ has a linear behavior on complete graphs of all sizes but under the particular circumstances of this experiment, it does look like VF2++ has a linear behavior. It's important to note that this may not be true on bigger graphs. Also, we can notice that for graphs and digraphs, the behavior of both algorithms is similar but in digraphs, the execution time is approximately twice that in graphs. This makes sense because in digraphs there are two types of edges to be considered in the matching function.
+
++++
+
+## Random Regular Graphs
+
++++
+
+A random $d$-regular graph is a graph randomly selected from all $d$-regular graphs of $n$ vertices, where $3\leq d < n$ and $nr$ is even. A $d$-regular graph is a graph where each node has exactly $d$ neighbors. That means that all nodes have the same degree. These graphs are interesting for our experiments because as we increase $d$ we will have graphs that get denser. This way we can easily test how the algorithms perform as we make graphs denser. In our example, we take a graph of 15 nodes and we will increase $d$. Again We will focus on positive instances.
+
+```{code-cell}
+vf2_all_times = []  # VF2 time measurements
+vf2pp_all_times = []  # VF2++ time measurements
+
+
+for i in range(0, 10, 2):
+    # vf2
+    vf2_time = np.median(
+        timeit.repeat(
+            setup="import networkx as nx;G = nx.random_regular_graph("
+            + str(i)
+            + ",15 , seed= 104)",
+            stmt="nx.is_isomorphic(G,G)",
+            timer=time.process_time,
+            number=1,
+            repeat=100,
+        )
+    )
+    vf2_all_times.append(vf2_time)
+
+    # vf2++
+    vf2pp_time = np.median(
+        timeit.repeat(
+            setup="import networkx as nx;G = nx.random_regular_graph("
+            + str(i)
+            + ",15 , seed= 104)",
+            stmt="nx.vf2pp_is_isomorphic(G,G)",
+            timer=time.process_time,
+            number=1,
+            repeat=100,
+        )
+    )
+    vf2pp_all_times.append(vf2pp_time)
+
+# Plot results
+fig, ax = plt.subplots()
+
+x = list(range(0, 10, 2))
+
+(vf2_line,) = ax.plot(x, vf2_all_times, label="vf2", linestyle="--", marker="o")
+(vf2pp_line,) = ax.plot(x, vf2pp_all_times, label="vf2++", linestyle="--", marker="o")
+
+plt.title("VF2 vs. VF2++ in random regular graphs of 15 nodes", fontweight="bold")
+plt.ylabel("Execution Time (seconds)")
+plt.xlabel("Number of neighbours")
+ax.legend(handles=[vf2_line, vf2pp_line])
+
+plt.show()
+```
+
+Once again, VF2++ is significantly faster than VF2. It's interesting to note that the execution time of VF2++ doesn't always increase as graphs become denser. This can be due to the fact that node ordering is more effective on some graphs than others. In some cases, unproductive matching functions may be explored first, causing VF2++ to be slower on smaller graphs.
+
++++
+
+## Erdos-Renyi Random graphs and digraphs
+A random erdos-renyi graph $G_{n, p}$ is a graph of $n$ nodes where all possible edges are added with probability $p$. With these graphs we can easily create graphs with similar density for multiple values of $n$. We will use them to see how the algorithms behave on graphs with the same density as we increase the number of nodes in positive instances.
+Let's do this experiment for different values of $p$:
+
+```{code-cell}
+p = [0.1, 0.3, 0.5, 0.6]  # Define probabilities
+
+vf2_all_times = []  # vf2 time measurements
+vf2pp_all_times = []  # vf2++ time measurements
+
+for j in range(0, len(p)):
+    vf2 = []
+    vf2pp = []
+    for i in range(0, 101, 10):
+        # vf2
+        vf2_time = np.median(
+            timeit.repeat(
+                setup="import networkx as nx;G = nx.erdos_renyi_graph("
+                + str(i)
+                + ","
+                + str(p[j])
+                + ", seed= 200)",
+                stmt="nx.is_isomorphic(G,G)",
+                timer=time.process_time,
+                number=1,
+                repeat=100,
+            )
+        )
+        vf2.append(vf2_time)
+
+        # vf2++
+        vf2pp_time = np.median(
+            timeit.repeat(
+                setup="import networkx as nx;G = nx.erdos_renyi_graph("
+                + str(i)
+                + ", "
+                + str(p[j])
+                + " , seed= 200)",
+                stmt="nx.vf2pp_is_isomorphic(G,G)",
+                timer=time.process_time,
+                number=1,
+                repeat=100,
+            )
+        )
+        vf2pp.append(vf2pp_time)
+    vf2_all_times.append(vf2)
+    vf2pp_all_times.append(vf2pp)
+
+# Plot results
+fig, axs = plt.subplots(2, 2)
+
+fig.set_size_inches(10, 5)
+
+x = list(range(0, 101, 10))
+
+plt.suptitle("VF2 vs. VF2++ in random erdos-renyi Graphs", fontweight="bold")
+
+(vf2_line1,) = axs[0][0].plot(
+    x, vf2_all_times[0], label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line1,) = axs[0][0].plot(
+    x, vf2pp_all_times[0], label="vf2++", linestyle="--", marker="o"
+)
+axs[0][0].set_title("p = 0.1")
+axs[0][0].set_ylabel("Execution Time (seconds)")
+axs[0][0].set_xlabel("Number of nodes")
+
+(vf2_line2,) = axs[0][1].plot(
+    x, vf2_all_times[1], label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line2,) = axs[0][1].plot(
+    x, vf2pp_all_times[1], label="vf2++", linestyle="--", marker="o"
+)
+axs[0][1].set_title("p = 0.3")
+axs[0][1].set_ylabel("Execution Time (seconds)")
+axs[0][1].set_xlabel("Number of nodes")
+
+(vf2_line2,) = axs[1][0].plot(
+    x, vf2_all_times[2], label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line2,) = axs[1][0].plot(
+    x, vf2pp_all_times[2], label="vf2++", linestyle="--", marker="o"
+)
+axs[1][0].set_title("p = 0.5")
+axs[1][0].set_ylabel("Execution Time (seconds)")
+axs[1][0].set_xlabel("Number of nodes")
+
+(vf2_line2,) = axs[1][1].plot(
+    x, vf2_all_times[3], label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line2,) = axs[1][1].plot(
+    x, vf2pp_all_times[3], label="vf2++", linestyle="--", marker="o"
+)
+axs[1][1].set_title("p = 0.6")
+axs[1][1].set_ylabel("Execution Time (seconds)")
+axs[1][1].set_xlabel("Number of nodes")
+
+axs[0][0].legend(handles=[vf2_line1, vf2pp_line1])
+axs[0][1].legend(handles=[vf2_line2, vf2pp_line2])
+axs[1][0].legend(handles=[vf2_line1, vf2pp_line1])
+axs[1][1].legend(handles=[vf2_line2, vf2pp_line2])
+
+fig.tight_layout(pad=2.0)
+plt.show()
+```
+
+Now we can do the same analysis in DiGraphs.
+
+```{code-cell}
+p = [0.1, 0.3, 0.5, 0.6]  # Define probabilities
+
+vf2_all_times = []  # VF2 time measurements
+vf2pp_all_times = []  # VF2++ time measurements
+
+for j in range(0, len(p)):
+    vf2 = []
+    vf2pp = []
+    for i in range(0, 101, 10):
+        # vf2
+        vf2_time = np.median(
+            timeit.repeat(
+                setup="import networkx as nx;G = nx.erdos_renyi_graph("
+                + str(i)
+                + ","
+                + str(p[j])
+                + ", seed= 200, directed=True)",
+                stmt="nx.is_isomorphic(G,G)",
+                timer=time.process_time,
+                number=1,
+                repeat=100,
+            )
+        )
+        vf2.append(vf2_time)
+
+        # vf2++
+        vf2pp_time = np.median(
+            timeit.repeat(
+                setup="import networkx as nx;G = nx.erdos_renyi_graph("
+                + str(i)
+                + ", "
+                + str(p[j])
+                + " , seed= 200, directed=True)",
+                stmt="nx.vf2pp_is_isomorphic(G,G)",
+                timer=time.process_time,
+                number=1,
+                repeat=100,
+            )
+        )
+        vf2pp.append(vf2pp_time)
+    vf2_all_times.append(vf2)
+    vf2pp_all_times.append(vf2pp)
+
+# Plot results
+fig, axs = plt.subplots(2, 2)
+
+fig.set_size_inches(10, 5)
+
+x = list(range(0, 101, 10))
+
+plt.suptitle("VF2 vs. VF2++ in random erdos-renyi DiGraphs", fontweight="bold")
+
+(vf2_line1,) = axs[0][0].plot(
+    x, vf2_all_times[0], label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line1,) = axs[0][0].plot(
+    x, vf2pp_all_times[0], label="vf2++", linestyle="--", marker="o"
+)
+axs[0][0].set_title("p = 0.1")
+axs[0][0].set_ylabel("Execution Time (seconds)")
+axs[0][0].set_xlabel("Number of nodes")
+
+(vf2_line2,) = axs[0][1].plot(
+    x, vf2_all_times[1], label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line2,) = axs[0][1].plot(
+    x, vf2pp_all_times[1], label="vf2++", linestyle="--", marker="o"
+)
+axs[0][1].set_title("p = 0.3")
+axs[0][1].set_ylabel("Execution Time (seconds)")
+axs[0][1].set_xlabel("Number of nodes")
+
+(vf2_line2,) = axs[1][0].plot(
+    x, vf2_all_times[2], label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line2,) = axs[1][0].plot(
+    x, vf2pp_all_times[2], label="vf2++", linestyle="--", marker="o"
+)
+axs[1][0].set_title("p = 0.5")
+axs[1][0].set_ylabel("Execution Time (seconds)")
+axs[1][0].set_xlabel("Number of nodes")
+
+(vf2_line2,) = axs[1][1].plot(
+    x, vf2_all_times[3], label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line2,) = axs[1][1].plot(
+    x, vf2pp_all_times[3], label="vf2++", linestyle="--", marker="o"
+)
+axs[1][1].set_title("p = 0.6")
+axs[1][1].set_ylabel("Execution Time (seconds)")
+axs[1][1].set_xlabel("Number of nodes")
+
+axs[0][0].legend(handles=[vf2_line1, vf2pp_line1])
+axs[0][1].legend(handles=[vf2_line2, vf2pp_line2])
+axs[1][0].legend(handles=[vf2_line1, vf2pp_line1])
+axs[1][1].legend(handles=[vf2_line2, vf2pp_line2])
+
+fig.tight_layout(pad=2.0)
+plt.show()
+```
+
+In both graphs and directed graphs, the experimental results are quite similar. VF2++ continues to outperform VF2. The growth of VF2's execution time increases exponentially as the number of nodes increases, while VF2++ has a linear behavior. Additionally, it's noteworthy that for all values of $p$, the curves of VF2 and VF2++ are very similar, with the only difference being the increase in execution time as the graphs become denser.
+
++++
+
+## Create graphs from a degree sequence
+
+To test on negative instances, in other words, graphs that are not isomorphic we need to create graphs with the same degree sequence. For that, we can use *nx.configuration_model(degree_sequence)* that creates a random MultiGraph with the given degree sequence. The degree sequence must have an even sum. Then we can just convert the resulting MultiGraph into a Graph. Let's see in an example how can we experiment with this:
+
+```{code-cell}
+# Example
+import random
+
+random.seed(11)  # Set a random seed
+d = random.choices(
+    list(range(0, 5)), k=11
+)  # Create a list with 11 random numbers in [0, 5]
+
+print("Is the sequence sum even?", np.sum(d) % 2 == 0)
+
+# Create 2 different random graphs with the same degree secuence
+g1 = nx.Graph(nx.configuration_model(d, seed=10))
+g1.remove_edges_from(nx.selfloop_edges(g1))
+
+g2 = nx.Graph(nx.configuration_model(d, seed=11))
+g2.remove_edges_from(nx.selfloop_edges(g2))
+
+print(g1)
+print("Are isomorphic? ", nx.is_isomorphic(g1, g2))
+```
+
+To do this same experiment on DiGraphs, we need to define an in-degree sequence and an out-degree sequence.  
+We will use the function *nx.directed_configuration_model(in_degree_sequence, out_degree_sequence, create_using, seed)*. Both degree sequences must have the same sum. Again this function returns a MultiDiGraph so we will have to convert it into a DiGraph. Let's do this in an example:
+
+```{code-cell}
+# Example
+import random
+
+random.seed(11)  # Set a random seed
+d = random.choices(
+    list(range(0, 5)), k=11
+)  # Create a list with 11 random numbers in [0, 5]
+
+print("Is the sequence sum even? ", np.sum(d) % 2 == 0)
+
+# Create 2 different random digraphs with the same degree secuence
+g1 = nx.directed_configuration_model(d, d, create_using=nx.DiGraph, seed=10)
+g1.remove_edges_from(nx.selfloop_edges(g1))
+
+g2 = nx.directed_configuration_model(d, d, create_using=nx.DiGraph, seed=11)
+g2.remove_edges_from(nx.selfloop_edges(g2))
+print(g1)
+print("Are isomorphic? ", nx.is_isomorphic(g1, g2))
+```
+
+Then we can create two random graphs with the same degree sequence and check that they are not isomorphic. The values of $k$ where chosen because they return a sequence with an even sum using these seeds.
+
+```{code-cell}
+k = [10, 11, 15, 18, 24, 32]  # Network sizes
+
+vf2_all_times_graphs = []  # VF2 time measurements on graphs
+vf2pp_all_times_graphs = []  # VF2++ time measurements on graphs
+
+vf2_all_times_digraphs = []  # VF2 time measurements on digraphs
+vf2pp_all_times_digraphs = []  # VF2++ time measurements on digraphs
+
+for i in range(0, len(k)):
+    # time vf2 on graphs
+    vf2_time_g = np.median(
+        timeit.repeat(
+            setup="import networkx as nx;import random;random.seed(11);d = random.choices(list(range(0, 5)), k ="
+            + str(k[i])
+            + ");g1 = nx.Graph(nx.configuration_model(d, seed = 11));g1.remove_edges_from(nx.selfloop_edges(g1)); g2 = nx.Graph(nx.configuration_model(d, seed = 11));g2.remove_edges_from(nx.selfloop_edges(g2))",
+            stmt="nx.is_isomorphic(g1,g2)",
+            timer=time.process_time,
+            number=1,
+            repeat=100,
+        )
+    )
+    vf2_all_times_graphs.append(vf2_time_g)
+
+    # time vf2++ on graphs
+    vf2pp_time_g = np.median(
+        timeit.repeat(
+            setup="import networkx as nx;import random;random.seed(11);d = random.choices(list(range(0, 5)), k ="
+            + str(k[i])
+            + ");g1 = nx.Graph(nx.configuration_model(d, seed = 11));g1.remove_edges_from(nx.selfloop_edges(g1)); g2 = nx.Graph(nx.configuration_model(d, seed = 11));g2.remove_edges_from(nx.selfloop_edges(g2))",
+            stmt="nx.is_isomorphic(g1,g2)",
+            timer=time.process_time,
+            number=1,
+            repeat=100,
+        )
+    )
+    vf2pp_all_times_graphs.append(vf2pp_time_g)
+
+    # time vf2 on digraphs
+    vf2_time_d = np.median(
+        timeit.repeat(
+            setup="import networkx as nx;import random;random.seed(11);d = random.choices(list(range(0, 5)), k ="
+            + str(k[i])
+            + ");g1 =nx.directed_configuration_model(d,d, create_using = nx.DiGraph, seed = 10);g1.remove_edges_from(nx.selfloop_edges(g1)); g2 =nx.directed_configuration_model(d,d, create_using = nx.DiGraph, seed = 11);g2.remove_edges_from(nx.selfloop_edges(g2))",
+            stmt="nx.is_isomorphic(g1,g2)",
+            timer=time.process_time,
+            number=1,
+            repeat=100,
+        )
+    )
+    vf2_all_times_digraphs.append(vf2_time_d)
+
+    # time vf2++ on digraphs
+    vf2pp_time_d = np.median(
+        timeit.repeat(
+            setup="import networkx as nx;import random;random.seed(11);d = random.choices(list(range(0, 5)), k ="
+            + str(k[i])
+            + ");g1 = nx.directed_configuration_model(d,d, create_using = nx.DiGraph, seed = 10);g1.remove_edges_from(nx.selfloop_edges(g1)); g2 = nx.directed_configuration_model(d,d, create_using = nx.DiGraph, seed = 11);g2.remove_edges_from(nx.selfloop_edges(g2))",
+            stmt="nx.is_isomorphic(g1,g2)",
+            timer=time.process_time,
+            number=1,
+            repeat=100,
+        )
+    )
+    vf2pp_all_times_digraphs.append(vf2pp_time_d)
+
+# Code used to generate the plot
+"""
+fig, ax = plt.subplots(1, 2)
+
+fig.set_size_inches(10, 5)
+
+plt.suptitle(
+    "VF2 vs. VF2++ in random Graphs and DiGraphs with the same degree secuence",
+    fontweight="bold",
+)
+
+# Plot vf2 and vf2++ in graphs
+
+(vf2_line_g,) = ax[0].plot(
+    k, vf2_all_times_graphs, label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line_g,) = ax[0].plot(
+    k, vf2pp_all_times_graphs, label="vf2++", linestyle="--", marker="o"
+)
+ax[0].set_title("Graphs")
+ax[0].set_ylabel("Execution Time (seconds)")
+ax[0].set_xlabel("Number of nodes")
+
+(vf2_line_d,) = ax[1].plot(
+    k, vf2_all_times_digraphs, label="vf2", linestyle="--", marker="o"
+)
+(vf2pp_line_d,) = ax[1].plot(
+    k, vf2pp_all_times_digraphs, label="vf2++", linestyle="--", marker="o"
+)
+ax[1].set_title("DiGraphs")
+ax[1].set_ylabel("Execution Time (seconds)")
+ax[1].set_xlabel("Number of nodes")
+
+ax[0].legend(handles=[vf2_line_g, vf2pp_line_g])
+ax[1].legend(handles=[vf2_line_d, vf2pp_line_d])
+
+fig.tight_layout(pad=3.0)
+"""
+# Show plot that was generated beforehand
+fig, ax = plt.subplots(1, figsize=(15, 10))
+img = plt.imread("same_degree_seq.png")
+
+ax.imshow(img)
+plt.axis("off")
+plt.show()
+```
+
+The results of the experiment on both graphs and directed graphs are quite similar. When the two graphs/directed graphs are not isomorphic, VF2 and VF2++ have similar performance. This outcome is understandable since both algorithms involve exploring all possible node matchings. However, it's also noteworthy that the enhancements in VF2++ do not negatively impact its performance even when all branches have to be partially explored. Additionally, it can be observed that the execution time on directed graphs is significantly higher than on graphs.
+
++++
+
+## Protein Dataset
+
++++
+
+To go one step further with our experiments we can test how the algorithms perform on specific datasets. We will work with a dataset of proteins. This dataset from TUDataset has 1113 graphs with node labels. The graphs have between 4 and 620 nodes.
+
+The dataset source is: <https://chrsmrrs.github.io/datasets/>
+
+The data is divided into 3 text files: 
+- PROTEINS_A: Edgelist of all graphs. 
+- PROTEINS_graph_indicator: Column vector of graph identifiers for all nodes of all graphs, the value in the i-th line is the graph_id of the node with node_id i. 
+- PROTEINS_node_attributes: The i-th line is the attribute of node i. 
+
+Let's see how can we read these files and generate NetworkX graphs.
+
+```{code-cell}
+import pandas as pd
+
+# Read edgelist
+edgelist = pd.read_csv(
+    "datasets/PROTEINS_A.txt", sep=", ", header=None, names=["node a", "node b"]
+)
+# Read graph_indicator
+graph_indicator = pd.read_csv(
+    "datasets/PROTEINS_graph_indicator.txt", names=["graph_ind"]
+)
+# Add the graph indicator into the edgelist dataframe
+edgelist["graph_indicator"] = list(graph_indicator["graph_ind"][edgelist["node a"] - 1])
+# Read node attributes
+node_att = list(
+    pd.read_csv("datasets/PROTEINS_node_attributes.txt", names=["node"])["node"]
+)
+
+graph_dataset = []
+for i in range(1, 1114):
+    # Create graph from edgelist
+    G = nx.from_pandas_edgelist(
+        edgelist[edgelist["graph_indicator"] == i], source="node a", target="node b"
+    )
+    # Add node attributes
+    for node in G.nodes:
+        G.nodes[node]["weight"] = node_att[node - 1]
+    graph_dataset.append(G)
+```
+
+Now that we have the dataset ready, we can go ahead and measure the execution time of both algorithms on the graphs. This takes some time so we can sort the graphs by size and run the algorithms in the n smaller graphs.
+
+```{code-cell}
+"""
+n = int(input("How many graphs do you want to test?"))
+reps = int(input("Enter the number of repetitions"))
+
+#Sort graphs by size 
+graph_sizes = [len(graph_dataset[i].nodes) for i in range(0, len(graph_dataset))]
+sizes_sorted = np.sort(graph_sizes)
+    
+order = np.argsort(graph_sizes)
+graph_db_sorted = np.array(graph_dataset, dtype=object)[order]
+
+#Measure time execution of VF2 and VF2++
+vf2_all_times_graphs = []  # VF2 time measurements on graphs
+vf2pp_all_times_graphs = []  # VF2++ time measurements on graphs
+
+for i in range(0,n):
+    G = graph_db_sorted[i]
+
+    med = np.array(np.zeros(reps))
+    for j in range(0, reps):
+        start = timeit.default_timer()
+
+        nx.is_isomorphic(G, G) #VF2
+
+        stop = timeit.default_timer()
+        total_time = stop - start
+        mins, secs = divmod(total_time, 60)
+        med[j] = secs 
+
+    vf2_all_times_graphs.append(np.median(med))
+
+    med = np.array(np.zeros(reps))
+    for j in range(0, reps):
+        start = timeit.default_timer()
+
+        nx.vf2pp_is_isomorphic(G, G) #VF2++
+
+        stop = timeit.default_timer()
+        total_time = stop - start
+        mins, secs = divmod(total_time, 60)
+        med[j] = secs 
+
+    vf2pp_all_times_graphs.append(np.median(med))
+
+#Create dataset with time measurements 
+p = pd.DataFrame()
+p["size"] = sizes_sorted[0:n]
+p["vf2"] = vf2_all_times_graphs
+p["vf2++"] = vf2pp_all_times_graphs
+
+#Calculate the median between all values by size 
+median = p.groupby(['size']).median() 
+
+# Plot experiment results 
+
+plt.plot(median["vf2"], label = "vf2")
+plt.plot(median["vf2++"], label = "vf2++")
+plt.legend(title= "Algorithms")
+plt.title("VF2 vs. VF2++ on PROTEINS dataset", fontweight="bold")
+plt.ylabel("Execution Time (seconds)")
+plt.xlabel("Number of nodes")
+plt.savefig('ex_PROTEINS.png')
+plt.show()
+"""
+```
+
+As the previous code takes too much time to run. Let's see the results of the experiment that were generated with the 900 smaller graphs and all measurements were repeated 300 times.
+
+```{code-cell}
+fig, ax = plt.subplots(1, figsize=(10, 10))
+img = plt.imread("ex_PROTEINS.png")
+plt.imshow(img)
+plt.axis("off")
+plt.show()
+```
+
+As we can see in the plot VF2++ performs significantly better than VF2 on the dataset. Also, the performance of both algorithms is the same as we saw in the previous experiments.
+
++++
+
+## Final conclusions
+After all the different experiments, We can conclude that
+- VF2++ performed significantly better than VF2 in all experiments on positive instances. But it's important to keep in mind that these experiments depend a lot on the state of the machine that was used. And to perform a proper benchmarking of the algorithms we should experiment with more graphs of different sizes.
+- The performance of VF2++ on the proteins dataset was significantly better than VF2 with similar results to the other experiments.
+- In negative instances the performance of both algorithms was similar which means that the improvements in VF2++ don't introduce a notable overhead.
+
++++
+
+## References
+- L. P. Cordella, P. Foggia, C. Sansone, M. Vento, “An Improved Algorithm for Matching Large Graphs”, 3rd IAPR-TC15 Workshop on Graph-based Representations in Pattern Recognition, Cuen, pp. 149-159, 2001. <https://www.researchgate.net/publication/200034365_An_Improved_Algorithm_for_Matching_Large_Graphs>
+- Jüttner, Alpár & Madarasi, Péter. (2018). “VF2++—An improved subgraph isomorphism algorithm”. Discrete Applied Mathematics. 242. <https://doi.org/10.1016/j.dam.2018.02.018>
diff --git a/content/exploratory_notebooks/facebook_notebook.md b/content/exploratory_notebooks/facebook_notebook.md
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     extension: .md
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diff --git a/content/generators/geometric.md b/content/generators/geometric.md
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@@ -5,7 +5,7 @@ jupytext:
     extension: .md
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+    jupytext_version: 1.14.2
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diff --git a/content/generators/index.md b/content/generators/index.md
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@@ -1,5 +1,24 @@
 
 
+
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+
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+
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+
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+
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+
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+
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+
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+
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+
 # Graph Generators
 
 A closer look at the functions provided by NetworkX to create interesting
diff --git a/content/generators/sudoku.md b/content/generators/sudoku.md
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     extension: .md
     format_name: myst
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+    jupytext_version: 1.14.2
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   display_name: Python 3 (ipykernel)
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