# Calculate the toughness of a graph

Let $G$ be a graph. **Toughness** is defined as
$$
t(G)=\min _S \frac{|S|}{c(G-S)}
$$
where $c$ is the number of connected components and the minimum is taken over all vertex cuts $S$ of $G$ (Graph toughness was first introduced by *Václav Chvátal* (1973)).

Based on the definition, I have written a code that relies mainly on third-party libraries `networkx`

and `itertools`

. I'm wondering if there is hope to improve its efficiency in the Sage or Python environment.

```
#Import the networkx library
import networkx as nx
# Import the combinations function from itertools
from itertools import combinations
def is_vertex_cut(G, vertex_cut):
# Create a copy of the graph
H = G.copy()
# Remove the nodes from the given set
H.remove_nodes_from(vertex_cut)
# Check the connectivity of the graph
return not nx.is_connected(H)
def toughness(G):
nodes = G.nodes()
order = G.number_of_nodes() - 2
vertex_conn = nx.node_connectivity(G)
min_toughness = float('inf') # Initialize the minimum toughness to positive infinity
for r in range(vertex_conn, order):
for j in combinations(nodes, r):
if is_vertex_cut(G, j):
# Create a copy of the graph
H = G.copy()
# Remove the nodes from the given set
H.remove_nodes_from(j)
# Calculate the number of connected components in (G-j)
num_connected_components = nx.number_connected_components(H)
toughness_value = r / num_connected_components
# Update the toughness value
min_toughness = min(min_toughness, toughness_value)
# Return the minimum toughness value
return min_toughness
# Test code
G = nx.cycle_graph(20)
# # G = nx.Graph()
# # G.add_nodes_from([1, 2, 3, 4, 5, 6])
# # G.add_edges_from([(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 1)])
min_toughness = toughness(G)
print("toughness:", min_toughness)
```

Then it gives us the following output:

toughness: 1.0

(run 166.5s)