Let
G
G
be an
n
n
-vertex graph with no minor isomorphic to an
h
h
-vertex complete graph. We prove that the vertices of
G
G
can be partitioned into three sets
A
,
B
,
C
A,\;B,\;C
such that no edge joins a vertex in
A
A
with a vertex in
B
B
, neither
A
A
nor
B
B
contains more than
2
n
/
3
2n/3
vertices, and
C
C
contains no more than
h
3
/
2
n
1
/
2
{h^{3/2}}{n^{1/2}}
vertices. This extends a theorem of Lipton and Tarjan for planar graphs. We exhibit an algorithm which finds such a partition
(
A
,
B
,
C
)
(A,\;B,\;C)
in time
O
(
h
1
/
2
n
1
/
2
m
)
O({h^{1/2}}{n^{1/2}}m)
, where
m
=
|
V
(
G
)
|
+
|
E
(
G
)
|
m = \left | {V(G)} \right | + \left | {E(G)} \right |
.