<s>
Consider	O
the	O
results	O
of	O
performing	O
a	O
depth-first	B-Algorithm
search	I-Algorithm
of	O
G	O
,	O
starting	O
at	O
any	O
vertex	O
,	O
and	O
assigning	O
each	O
vertex	O
v	O
to	O
a	O
set	O
Vi	O
where	O
i	O
is	O
the	O
length	O
(	O
taken	O
mod	O
k	O
)	O
of	O
the	O
path	O
in	O
the	O
depth-first	B-Algorithm
search	I-Algorithm
tree	O
from	O
the	O
root	O
to	O
v	O
.	O
It	O
can	O
be	O
shown	O
that	O
this	O
partition	O
into	O
sets	O
Vi	O
has	O
the	O
property	O
that	O
each	O
edge	O
in	O
the	O
graph	O
goes	O
from	O
a	O
set	O
Vi	O
to	O
another	O
set	O
V( i	O
+	O
1	O
)	O
mod	O
k	O
.	O
Conversely	O
,	O
if	O
a	O
partition	O
with	O
this	O
property	O
exists	O
for	O
a	O
strongly	O
connected	O
graph	O
G	O
,	O
k	O
must	O
divide	O
the	O
lengths	O
of	O
all	O
cycles	O
in	O
G	O
.	O
</s>
<s>
For	O
each	O
e	O
in	O
G	O
that	O
connects	O
a	O
vertex	O
on	O
level	O
i	O
of	O
the	O
depth-first	B-Algorithm
search	I-Algorithm
tree	O
to	O
a	O
vertex	O
on	O
level	O
j	O
,	O
let	O
ke	O
=	O
j	O
-	O
i	O
-	O
1	O
.	O
</s>
<s>
Jarvis	O
and	O
Shier	O
describe	O
a	O
very	O
similar	O
algorithm	O
using	O
breadth	B-Algorithm
first	I-Algorithm
search	I-Algorithm
in	O
place	O
of	O
depth-first	B-Algorithm
search	I-Algorithm
;	O
the	O
advantage	O
of	O
depth-first	B-Algorithm
search	I-Algorithm
is	O
that	O
the	O
strong	O
connectivity	O
analysis	O
can	O
be	O
incorporated	O
into	O
the	O
same	O
search	O
.	O
</s>
