<s>
The	O
Kleitman	B-Algorithm
–	I-Algorithm
Wang	I-Algorithm
algorithms	I-Algorithm
are	O
two	O
different	O
algorithms	O
in	O
graph	O
theory	O
solving	O
the	O
digraph	O
realization	O
problem	O
,	O
i.e.	O
</s>
<s>
the	O
question	O
if	O
there	O
exists	O
for	O
a	O
finite	O
list	B-Language
of	O
nonnegative	O
integer	O
pairs	O
a	O
simple	O
directed	O
graph	O
such	O
that	O
its	O
degree	O
sequence	O
is	O
exactly	O
this	O
list	B-Language
.	O
</s>
<s>
For	O
a	O
positive	O
answer	O
the	O
list	B-Language
of	O
integer	O
pairs	O
is	O
called	O
digraphic	O
.	O
</s>
<s>
Let	O
be	O
a	O
finite	O
list	B-Language
of	O
nonnegative	O
integers	O
that	O
is	O
in	O
nonincreasing	O
lexicographical	O
order	O
and	O
let	O
be	O
a	O
pair	O
of	O
nonnegative	O
integers	O
with	O
.	O
</s>
<s>
List	B-Language
is	O
digraphic	O
if	O
and	O
only	O
if	O
the	O
finite	O
list	B-Language
has	O
nonnegative	O
integer	O
pairs	O
and	O
is	O
digraphic	O
.	O
</s>
<s>
If	O
the	O
given	O
list	B-Language
digraphic	O
then	O
the	O
theorem	O
will	O
be	O
applied	O
at	O
most	O
times	O
setting	O
in	O
each	O
further	O
step	O
.	O
</s>
<s>
This	O
process	O
ends	O
when	O
the	O
whole	O
list	B-Language
consists	O
of	O
pairs	O
.	O
</s>
<s>
if	O
it	O
is	O
possible	O
to	O
reduce	O
the	O
list	B-Language
to	O
,	O
then	O
we	O
add	O
arcs	O
.	O
</s>
<s>
When	O
the	O
list	B-Language
cannot	O
be	O
reduced	O
to	O
a	O
list	B-Language
of	O
nonnegative	O
integer	O
pairs	O
in	O
any	O
step	O
of	O
this	O
approach	O
,	O
the	O
theorem	O
proves	O
that	O
the	O
list	B-Language
from	O
the	O
beginning	O
is	O
not	O
digraphic	O
.	O
</s>
<s>
Let	O
be	O
a	O
finite	O
list	B-Language
of	O
nonnegative	O
integers	O
such	O
that	O
and	O
let	O
be	O
a	O
pair	O
such	O
that	O
is	O
maximal	O
with	O
respect	O
to	O
the	O
lexicographical	O
order	O
under	O
all	O
pairs	O
.	O
</s>
<s>
List	B-Language
is	O
digraphic	O
if	O
and	O
only	O
if	O
the	O
finite	O
list	B-Language
has	O
nonnegative	O
integer	O
pairs	O
and	O
is	O
digraphic	O
.	O
</s>
<s>
Note	O
that	O
the	O
list	B-Language
must	O
not	O
be	O
in	O
lexicographical	O
order	O
as	O
in	O
the	O
first	O
version	O
.	O
</s>
<s>
If	O
the	O
given	O
list	B-Language
is	O
digraphic	O
,	O
then	O
the	O
theorem	O
will	O
be	O
applied	O
at	O
most	O
times	O
,	O
setting	O
in	O
each	O
further	O
step	O
.	O
</s>
<s>
This	O
process	O
ends	O
when	O
the	O
whole	O
list	B-Language
consists	O
of	O
pairs	O
.	O
</s>
<s>
if	O
it	O
is	O
possible	O
to	O
reduce	O
the	O
list	B-Language
to	O
,	O
then	O
one	O
adds	O
arcs	O
.	O
</s>
<s>
When	O
the	O
list	B-Language
cannot	O
be	O
reduced	O
to	O
a	O
list	B-Language
of	O
nonnegative	O
integer	O
pairs	O
in	O
any	O
step	O
of	O
this	O
approach	O
,	O
the	O
theorem	O
proves	O
that	O
the	O
list	B-Language
from	O
the	O
beginning	O
is	O
not	O
digraphic	O
.	O
</s>
