<s>
In	O
graph	O
theory	O
,	O
a	O
rainbow-independent	B-Algorithm
set	I-Algorithm
(	O
ISR	O
)	O
is	O
an	O
independent	O
set	O
in	O
a	O
graph	O
,	O
in	O
which	O
each	O
vertex	O
has	O
a	O
different	O
color	O
.	O
</s>
<s>
A	O
set	O
of	O
vertices	O
is	O
called	O
a	O
rainbow-independent	B-Algorithm
set	I-Algorithm
if	O
it	O
satisfies	O
both	O
the	O
following	O
conditions	O
:	O
</s>
<s>
Other	O
terms	O
used	O
in	O
the	O
literature	O
are	O
independent	B-Algorithm
set	I-Algorithm
of	I-Algorithm
representatives	I-Algorithm
,	O
independent	B-Algorithm
transversal	I-Algorithm
,	O
and	O
independent	B-Algorithm
system	I-Algorithm
of	I-Algorithm
representatives	I-Algorithm
.	O
</s>
<s>
It	O
is	O
known	O
that	O
a	O
single	O
simplex	O
is	O
-connected	O
for	O
all	O
integers	O
,	O
since	O
all	O
its	O
reduced	O
homology	O
groups	O
are	O
trivial	O
(	O
see	O
simplicial	B-Algorithm
homology	I-Algorithm
)	O
.	O
</s>
<s>
Every	O
properly	O
coloured	O
triangle-free	O
graph	O
of	O
chromatic	O
number	O
contains	O
a	O
rainbow-independent	B-Algorithm
set	I-Algorithm
of	O
size	O
at	O
least	O
.	O
</s>
<s>
Several	O
authors	O
have	O
studied	O
conditions	O
for	O
existence	O
of	O
large	O
rainbow-independent	B-Algorithm
sets	I-Algorithm
in	O
various	O
classes	O
of	O
graphs	O
.	O
</s>
<s>
The	O
ISR	O
decision	O
problem	O
is	O
the	O
problem	O
of	O
deciding	O
whether	O
a	O
given	O
graph	O
and	O
a	O
given	O
partition	O
of	O
into	O
colors	O
admits	O
a	O
rainbow-independent	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
An	O
alternative	O
proof	O
is	O
by	O
reduction	O
from	O
SAT	B-Algorithm
.	O
</s>
<s>
Hence	O
,	O
a	O
rainbow-independent	B-Algorithm
set	I-Algorithm
in	O
is	O
a	O
rainbow	O
matching	O
in	O
.	O
</s>
<s>
An	O
independent	B-Algorithm
transversal	I-Algorithm
is	O
a	O
committee	O
with	O
no	O
conflict	O
,	O
with	O
exactly	O
one	O
member	O
from	O
each	O
department	O
.	O
</s>
<s>
Therefore	O
,	O
every	O
rainbow	O
clique	O
in	O
a	O
graph	O
corresponds	O
to	O
a	O
rainbow-independent	B-Algorithm
set	I-Algorithm
in	O
its	O
complement	O
graph	O
.	O
</s>
