# Create program to find which graphs contain specific subgraph

Hello, I am pretty new to Sage, and am discovering all the wonderful things it can do. I know how to ask Sage to generate a list of graphs that satisfy particular properties. To do this I "call" a program named "nauty". For example, if I would like to generate all graphs with 10 vertices, 25 edges, and clique number 4 I type the following:

sage: g10=[g for g in graphs.nauty_geng('10 25') if g.clique_number()==4]

Now say that I want to check which of these graphs contain a particular subgraph. For example, say I want to know which one of these graphs contain a 5-cycle. How can I ask Sage to go through the list of graphs in g10 and tell me this information?

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g10=[g for g in graphs.nauty_geng('10 25') if g.clique_number()==4]
H = graphs.CycleGraph(5)
for g in g10:
if g.subgraph_search(H,induced=False):
g.show()


After defining the subgraph you want, the cycle on 5 vertices, look through the graphs in g10 and figure out if it has H, not necessarily induced. It then prints out the graphs. The documentation for subgraph_search is here.

EDIT: With respect to your comment below, I think that it is possible but I don't know enough of the intricacies of the graph theory commands. Consider this code.

g10=[g for g in graphs.nauty_geng('10 25') if g.clique_number()==4]
H = graphs.CycleGraph(5)
K = graphs.CompleteGraph(4)
for g in g10:
if g.subgraph_search(H,induced=False):
for p in g.subgraph_search_iterator(K,induced=True):
for v in p:
if g.degree(v)<6:
print "False"
print p
g.show()
else:
print "True"


For each graph in g10 it looks for a non-induced C_5. If the graph has it then it looks for an induced K_4. When it finds it, it looks at all the vertices in it (the subgraph p which is a K_4) and looks to see if the degree of any vertex is less than 6. If so, it prints out "False", the 4 vertices inducing K_4 and the graph from g10 that they are a subgraph of. If you didn't require the graph to have H then you can easily modify the code to what you want.

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Thank you for your help. Would you also happen to know how I can ask Sage to give me graphs in g10 that also satisfy the condition that every vertex in a 4-clique has degree at least 6?

( 2019-03-31 20:02:03 +0200 )edit

Thank you!! You are a lifesaver! I was unaware of subgraph_search_iterator tool, this is wonderful.

( 2019-04-01 04:14:58 +0200 )edit

There is an extensive, almost 900 page, pdf reference doc just on graph theory which is available here.

( 2019-04-01 04:51:57 +0200 )edit