2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 4550
author:  Bruce Reed and David R. Wood 

title:  Fast separation in a graph with an excluded minor 
keywords:  graph algorithm, separator, minor 
abstract: 
Let
G
be an
n
vertex
m
edge graph with weighted vertices. A pair of vertex
sets
A,B⊆V(G)
is a 2/3separation of order
A∩B
if
A ∪B = V(G)
, there is no edge between
A \ B
and
B \ A
, and both
A \ B
and
B \ A
have weight at most
2/3
the total weight of
G
. Let
ℓ∈ℤ+
be fixed. Alon, Seymour and Thomas
[J. Amer. Math. Soc. 1990]
presented an algorithm that in
O
(n
1/2
m)
K
minor of
ℓ
G
, or a separation of
G
of order
O
(n
1/2
)
O
(n+m)
O
(n+m)
O
(n
2/3
)
ε∈[0,1/2]
, our algorithm either outputs a
K
minor of
ℓ
G
, or a separation of
G
with order
O
(n
(2ε)/3
)
O
(n
1+ε
+m)

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reference:  Bruce Reed and David R. Wood (2005), Fast separation in a graph with an excluded minor , in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 4550 
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