<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
Bron	B-Algorithm
–	I-Algorithm
Kerbosch	I-Algorithm
algorithm	I-Algorithm
is	O
an	O
enumeration	B-Algorithm
algorithm	I-Algorithm
for	O
finding	O
all	O
maximal	O
cliques	O
in	O
an	O
undirected	O
graph	O
.	O
</s>
<s>
The	O
Bron	B-Algorithm
–	I-Algorithm
Kerbosch	I-Algorithm
algorithm	I-Algorithm
was	O
designed	O
by	O
Dutch	O
scientists	O
Coenraad	O
Bron	O
and	O
Joep	O
Kerbosch	O
,	O
who	O
published	O
its	O
description	O
in	O
1973	O
.	O
</s>
<s>
Although	O
other	O
algorithms	O
for	O
solving	O
the	O
clique	O
problem	O
have	O
running	O
times	O
that	O
are	O
,	O
in	O
theory	O
,	O
better	O
on	O
inputs	O
that	O
have	O
few	O
maximal	O
independent	O
sets	O
,	O
the	O
Bron	B-Algorithm
–	I-Algorithm
Kerbosch	I-Algorithm
algorithm	I-Algorithm
and	O
subsequent	O
improvements	O
to	O
it	O
are	O
frequently	O
reported	O
as	O
being	O
more	O
efficient	O
in	O
practice	O
than	O
the	O
alternatives	O
.	O
</s>
<s>
A	O
contemporaneous	O
algorithm	O
of	O
,	O
although	O
presented	O
in	O
different	O
terms	O
,	O
can	O
be	O
viewed	O
as	O
being	O
the	O
same	O
as	O
the	O
Bron	B-Algorithm
–	I-Algorithm
Kerbosch	I-Algorithm
algorithm	I-Algorithm
,	O
as	O
it	O
generates	O
the	O
same	O
search	O
tree	O
.	O
</s>
<s>
The	O
basic	O
form	O
of	O
the	O
Bron	B-Algorithm
–	I-Algorithm
Kerbosch	I-Algorithm
algorithm	I-Algorithm
is	O
a	O
recursive	O
backtracking	B-Algorithm
algorithm	I-Algorithm
that	O
searches	O
for	O
all	O
maximal	O
cliques	O
in	O
a	O
given	O
graph	O
G	O
.	O
More	O
generally	O
,	O
given	O
three	O
disjoint	O
sets	O
of	O
vertices	O
R	O
,	O
P	O
,	O
and	O
X	O
,	O
it	O
finds	O
the	O
maximal	O
cliques	O
that	O
include	O
all	O
of	O
the	O
vertices	O
in	O
R	O
,	O
some	O
of	O
the	O
vertices	O
in	O
P	O
,	O
and	O
none	O
of	O
the	O
vertices	O
in	O
X	O
.	O
</s>
<s>
An	O
alternative	O
method	O
for	O
improving	O
the	O
basic	O
form	O
of	O
the	O
Bron	B-Algorithm
–	I-Algorithm
Kerbosch	I-Algorithm
algorithm	I-Algorithm
involves	O
forgoing	O
pivoting	O
at	O
the	O
outermost	O
level	O
of	O
recursion	O
,	O
and	O
instead	O
choosing	O
the	O
ordering	O
of	O
the	O
recursive	O
calls	O
carefully	O
in	O
order	O
to	O
minimize	O
the	O
sizes	O
of	O
the	O
sets	O
of	O
candidate	O
vertices	O
within	O
each	O
recursive	O
call	O
.	O
</s>
<s>
The	O
degeneracy	B-Algorithm
of	O
a	O
graph	O
is	O
the	O
smallest	O
number	O
such	O
that	O
every	O
subgraph	O
of	O
has	O
a	O
vertex	O
with	O
degree	O
or	O
less	O
.	O
</s>
<s>
Every	O
graph	O
has	O
a	O
degeneracy	B-Algorithm
ordering	O
,	O
an	O
ordering	O
of	O
the	O
vertices	O
such	O
that	O
each	O
vertex	O
has	O
or	O
fewer	O
neighbors	O
that	O
come	O
later	O
in	O
the	O
ordering	O
;	O
a	O
degeneracy	B-Algorithm
ordering	O
may	O
be	O
found	O
in	O
linear	O
time	O
by	O
repeatedly	O
selecting	O
the	O
vertex	O
of	O
minimum	O
degree	O
among	O
the	O
remaining	O
vertices	O
.	O
</s>
<s>
If	O
the	O
order	O
of	O
the	O
vertices	O
that	O
the	O
Bron	B-Algorithm
–	I-Algorithm
Kerbosch	I-Algorithm
algorithm	I-Algorithm
loops	O
through	O
is	O
a	O
degeneracy	B-Algorithm
ordering	O
,	O
then	O
the	O
set	O
of	O
candidate	O
vertices	O
in	O
each	O
call	O
(	O
the	O
neighbors	O
of	O
that	O
are	O
later	O
in	O
the	O
ordering	O
)	O
will	O
be	O
guaranteed	O
to	O
have	O
size	O
at	O
most	O
.	O
</s>
<s>
This	O
variant	O
of	O
the	O
algorithm	O
can	O
be	O
proven	O
to	O
be	O
efficient	O
for	O
graphs	O
of	O
small	O
degeneracy	B-Algorithm
,	O
and	O
experiments	O
show	O
that	O
it	O
also	O
works	O
well	O
in	O
practice	O
for	O
large	O
sparse	O
social	O
networks	O
and	O
other	O
real-world	O
graphs	O
.	O
</s>
<s>
The	O
graph	O
in	O
the	O
example	O
has	O
degeneracy	B-Algorithm
two	O
;	O
one	O
possible	O
degeneracy	B-Algorithm
ordering	O
is	O
6	O
,	O
4	O
,	O
3	O
,	O
1	O
,	O
2	O
,	O
5	O
.	O
</s>
<s>
The	O
Bron	B-Algorithm
–	I-Algorithm
Kerbosch	I-Algorithm
algorithm	I-Algorithm
is	O
not	O
an	O
output-sensitive	B-Language
algorithm	I-Language
:	O
unlike	O
some	O
other	O
algorithms	O
for	O
the	O
clique	O
problem	O
,	O
it	O
does	O
not	O
run	O
in	O
polynomial	O
time	O
per	O
maximal	O
clique	O
generated	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
efficient	O
in	O
a	O
worst-case	O
sense	O
:	O
by	O
a	O
result	O
of	O
,	O
any	O
n-vertex	O
graph	O
has	O
at	O
most	O
3n/3	O
maximal	O
cliques	O
,	O
and	O
the	O
worst-case	O
running	O
time	O
of	O
the	O
Bron	B-Algorithm
–	I-Algorithm
Kerbosch	I-Algorithm
algorithm	I-Algorithm
(	O
with	O
a	O
pivot	O
strategy	O
that	O
minimizes	O
the	O
number	O
of	O
recursive	O
calls	O
made	O
at	O
each	O
step	O
)	O
is	O
O( 	O
3n/3	O
)	O
,	O
matching	O
this	O
bound	O
.	O
</s>
<s>
In	O
particular	O
the	O
vertex-ordering	O
version	O
of	O
the	O
Bron	B-Algorithm
–	I-Algorithm
Kerbosch	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
made	O
to	O
run	O
in	O
time	O
,	O
where	O
is	O
the	O
degeneracy	B-Algorithm
of	O
the	O
graph	O
,	O
a	O
measure	O
of	O
its	O
sparseness	O
.	O
</s>
