<s>
In	O
theoretical	O
computer	O
science	O
,	O
a	O
certifying	B-Error_Name
algorithm	I-Error_Name
is	O
an	O
algorithm	O
that	O
outputs	O
,	O
together	O
with	O
a	O
solution	O
to	O
the	O
problem	O
it	O
solves	O
,	O
a	O
proof	O
that	O
the	O
solution	O
is	O
correct	O
.	O
</s>
<s>
A	O
certifying	B-Error_Name
algorithm	I-Error_Name
is	O
said	O
to	O
be	O
efficient	O
if	O
the	O
combined	O
runtime	O
of	O
the	O
algorithm	O
and	O
a	O
proof	O
checker	O
is	O
slower	O
by	O
at	O
most	O
a	O
constant	O
factor	O
than	O
the	O
best	O
known	O
non-certifying	O
algorithm	O
for	O
the	O
same	O
problem	O
.	O
</s>
<s>
The	O
proof	O
produced	O
by	O
a	O
certifying	B-Error_Name
algorithm	I-Error_Name
should	O
be	O
in	O
some	O
sense	O
simpler	O
than	O
the	O
algorithm	O
itself	O
,	O
for	O
otherwise	O
any	O
algorithm	O
could	O
be	O
considered	O
certifying	O
(	O
with	O
its	O
output	O
verified	O
by	O
running	O
the	O
same	O
algorithm	O
again	O
)	O
.	O
</s>
<s>
Implementations	O
of	O
certifying	B-Error_Name
algorithms	I-Error_Name
that	O
also	O
include	O
a	O
checker	O
for	O
the	O
proof	O
generated	O
by	O
the	O
algorithm	O
may	O
be	O
considered	O
to	O
be	O
more	O
reliable	O
than	O
non-certifying	O
algorithms	O
.	O
</s>
<s>
In	O
contrast	O
,	O
a	O
certifying	B-Error_Name
algorithm	I-Error_Name
might	O
output	O
a	O
2-coloring	O
of	O
the	O
graph	O
in	O
the	O
case	O
that	O
it	O
is	O
bipartite	O
,	O
or	O
a	O
cycle	O
of	O
odd	O
length	O
if	O
it	O
is	O
not	O
.	O
</s>
<s>
Analogously	O
,	O
it	O
is	O
possible	O
to	O
test	O
whether	O
a	O
given	O
directed	O
graph	O
is	O
acyclic	O
by	O
a	O
certifying	B-Error_Name
algorithm	I-Error_Name
that	O
outputs	O
either	O
a	O
topological	B-Algorithm
order	I-Algorithm
or	O
a	O
directed	O
cycle	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
test	O
whether	O
an	O
undirected	O
graph	O
is	O
a	O
chordal	O
graph	O
by	O
a	O
certifying	B-Error_Name
algorithm	I-Error_Name
that	O
outputs	O
either	O
an	O
elimination	O
ordering	O
(	O
an	O
ordering	O
of	O
all	O
vertices	O
such	O
that	O
,	O
for	O
every	O
vertex	O
,	O
the	O
neighbors	O
that	O
are	O
later	O
in	O
the	O
ordering	O
form	O
a	O
clique	O
)	O
or	O
a	O
chordless	O
cycle	O
.	O
</s>
<s>
And	O
it	O
is	O
possible	O
to	O
test	O
whether	O
a	O
graph	O
is	O
planar	O
by	O
a	O
certifying	B-Error_Name
algorithm	I-Error_Name
that	O
outputs	O
either	O
a	O
planar	O
embedding	O
or	O
a	O
Kuratowski	O
subgraph	O
.	O
</s>
