MATH10200705001 Neeldhara Misra - Homi Bhabha National ...

KERNELS FOR THE F-DELETION PROBLEM
By
Neeldhara Misra
The Institute of Mathematical Sciences, Chennai.
Athesissubmittedtothe
Board of Studies in Mathematical Sciences
In partial fulfillment of the requirements
For the Degree of
DOCTOR OF PHILOSOPHY
of
HOMI BHABHA NATIONAL INSTITUTE
September 2011

Homi Bhabha National Institute
Recommendations of the Viva Voce Board
As members of the Viva Voce Board, we recommend that the dissertation prepared
by Neeldhara Misra entitled “Kernels for the F-Deletion Problem” may
be accepted as fulfilling the dissertation requirement for the Degree of Doctor of
Philosophy.
Chairman : Chair of committe
Convener : Conv of Committe
Member : Member 1 of committe
Member : Member 2 of committe
Date :
Date :
Date :
Date :
Final approval and acceptance of this dissertation is contingent upon the candidate’s
submission of the final copies of the dissertation to HBNI.
IherebycertifythatIhavereadthisdissertationpreparedundermydirection
and recommend that it may be accepted as fulfilling the dissertation requirement.
Guide : Venkatesh Raman
Date :

DECLARATION
I, hereby declare that the investigation presented in the thesis
has been carried out by me. The work is original and the
work has not been submitted earlier as a whole or in part for
adegree/diplomaatthisoranyotherInstitutionorUniversity.
Neeldhara Misra

ACKNOWLEDGEMENTS
Ihavebeenextremelyfortunateinhavingenjoyedtheguidanceandcareofmy
advisor, Prof. Venkatesh Raman. His presence has been a source of inspiration and
strength like no other, and every opportunity of interaction has been a treasured
learning experience. More importantly than leading me up to successes, he found
me the motivation that I needed to last me through the inevitably larger number
of failures. I will never cease be amazed at his expertise and teaching, and my own
stroke of luck for having had the chance to witness it from close quarters.
I am also deeply obliged to Prof. Saket Saurabh for his relentless efforts in
advancing my comprehension of a number of things, technical and non-technical
alike. He has always been incredibly accessible and it is thanks to his infinite
patience that I eventually got around to messing with things that I would otherwise
keep at a safe distance. I also learned from him to value the pursuit as much as, and
irrespective of, the associated discovery. The energy and intensity that he brings
to research is nothing short of magical, and the experience has been a privilege.
Prof. Mike Fellows and Frances Rosamond have my eternal gratitude for taking
care of me at various stages, and ensuring that I have nothing less than the time
of my life! They have been very generous in their enthusiastic support, and I have
certainly received from them much more than I deserve.
I would like to thank Prof. Fedor Fomin, whom I had an opportunity to
visit at the University of Bergen — the stay was intensely enriching. He was
very generous in with sharing his profound experience, and the discussions were
delightful. Special thanks are due to Prof. Pinar Heggernes, Pim van’t Hof, Daniel
Marx, and Yngve Villanger for a particularly memorable collaboration. I learned
agooddealfromDanielLokshtanov,andIamthankfulforhistimeandvaluable
insights.
Thanks to Somnath Sikdar, Geevarghese Philip, and M. S. Ramanujan, the
typical day was always something to look forward to!
I would like to thank Abhimanyu M. Ambalath, Radheshyam Balasundaram,
Chintan Rao H., Venkata Koppula, Matthias Mnich, N. S. Narayanaswamy, and
Bal Sri Shankar, for many exciting discussions. I also enjoyed very much a number
of discussions with Prof. Ronojoy Adhikari - thank you!

I will be forever grateful to all the faculty at the Department of Theoretical
Computer Science. Any understanding that I might have developed about Computer
Science certainly is thanks to their patient, creative, and inspiring teaching. I
am equally thankful to all my friends and colleagues for being tirelessly supportive.
The work leading to this thesis executed at IMSc and the University of Bergen. I
would like to thank both institutions for the financial support, excellent infrastructure
and wonderful research environments. Special thanks to the administrative
staff, who have always been especially helpful.
My parents happen to be my life, universe, and everything. For chasing my
dreams with me, in true no-matter-what style, thanks very much indeed. I will
add that more can be said that goes without saying, and is therefore left unsaid.
5

Abstract
In this thesis, we use the parameterized framework for the design and analysis
of algorithms for NP-complete problems. This amounts to studying the parameterized
version of the classical decision version. Herein, the classical language
appended with a secondary measure called a “parameter”. The central notion in
parameterized complexity is that of fixed-parameter tractability, which means given
an instance (x, k) of a parameterized language L, deciding whether (x, k) ∈ L in
time f(k) · p(|x|), where f is an arbitrary function of k alone and p is a polynomial
function. The notion of kernelization formalizes preprocessing or data reduction,
and refers to polynomial time algorithms that transform any given input into an
equivalent instance whose size is bounded as a function of the parameter alone.
The center of our attention in this thesis is the F-Deletion problem, a vastly
general question that encompasses many fundamental optimization problems as
special cases. In particular, we provide evidence supporting a conjecture about the
kernelization complexity of the problem, and this work branches off in a number of
directions, leading to results of independent interest. We also study the Colorful
Motifs problem, a well-known question that arises frequently in practice. Our
investigation demonstrates the hardness of the problem even when restricted to
very simple graph classes.
The F-Deletion Problem Let F be a finite family of graphs. The F-Deletion
problem takes as input a graph G on n vertices, and a positive integer k. The
question is whether it is possible to delete at most k vertices from G such that
the remaining graph contains no graph from F as a minor. This question encompasses
fundamental problems such as Vertex Cover (consider F consisting of
the graph with two vertices and one edge) or Feedback Vertex Set (the set
F consists of a cycle with three vertices). A number of other deletion-based optimization
problems also turn out to be special cases of Planar F-Deletion, for
instance: Diamond Hitting Set, Outerplanar Deletion Set, Pathwidth
r-Deletion Set and Treewidth r-Deletion Set.
The general F-Deletion problem is NP-complete. From the parameterized
perspective, by one of the most well-known consequences of the celebrated Graph
Minor theory of Robertson and Seymour, the F-Deletion problem is fixed-
6

parameter tractable for every finite set of forbidden minors. It is conjectured that
the F-Deletion problem admits a polynomial kernel if, and only if, F contains
aplanargraph. WerefertotheF-Deletion question when restricted to the case
when F contains a planar graph as the Planar F-Deletion problem.
We obtain a number of kernelization results for the Planar F-Deletion problem.
We give a linear vertex kernel on graphs excluding t-claw K 1,t , the star with
t leves, as an induced subgraph, where t is a fixed integer, and a obtain polynomial
kernels for the case when F contains graph θ c as a minor for a fixed integer
c. The graph θ c consists of two vertices connected by c parallel edges. Even
though this may appear to be a very restricted class of problems it already encompasses
well-studied problems such as Vertex Cover, Feedback Vertex Set
and Diamond Hitting Set. The generic kernelization algorithm is based on a
non-trivial application of protrusion techniques, previously used only for problems
on topological graph classes. We also demonstrate an approximation algorithm
achieving an approximation ratio of O(log 3/2 OPT), where OPT is the size of an
optimal solution on general undirected graphs. The approximation algorithm is
used crucially in the kernelization routines.
We also show an algorithm for Planar F-Deletion that runs in time 2 O(k log k) n 2 ,
improving substantially from what was known previously, 2 2O(k log k) n O(1) . Since
Planar F-Deletion is a generalization of Vertex Cover, wehavethatitcannot
be solved in time 2 o(k) n O(1) , unless Exponential Time Hypothesis fails, so our
algorithm is also quite close to being optimal. This algorithm uses the technique
of iterative compression, and the instance encountered at every iteration is subject
to kernelization. The intermediate kernelization algorithm is quite non-trivial and
requires reformulating some of the fundamental machinery that was used in all the
other situations. Also, towards the kernel, we show a “decomposition lemma” that
asserts that any graph of constant treewidth can essentially be partitioned into a
number of protrusions. We believe this result to be of independent interest.
We also consider the complementary “packing” question: we wish to maximize
the number of vertex (or edge) disjoint minor models of graphs in F. We show
that the packing number is closely related to the size of the optimal F-hitting
set in the special case when F = {θ c }. Independently, but on a related note, we
show two lower bounds, demonstrating that the problem of finding if there are at
least k vertex-disjoint minor models of θ c is unlikely to admit a polynomial kernel
7

parameterized by k, andthisisalsotruefortheproblemofcheckingifthereare
at least k vertex-disjoint cycles of odd length (again parameterized by k).
Colorful Motifs We study the problem of Colorful Motifs on various graph
classes. We prove that the problem of Colorful Motifs restricted to superstars
is NP-complete. Further, we show NP-completeness on graphs of diameter two.
We apply this result towards settling the classical complexity of Connected
Dominating Set on graphs of diameter two — specifically, we show that it is
NP-complete. Further, we show that on graphs of diameter two, the problem is
NP-complete and is unlikely to admit a polynomial kernel.
Next, we show that obtaining polynomial kernels for Colorful Motifs on
comb graphs is infeasible, but we show the existence of n polynomial kernels.
Further, we study the problem of Colorful Motifs on trees, where we observe
that the natural strategies for many polynomial kernels are not successful. For
instance, we show that “guessing” a root vertex, which helped in the case of comb
graphs, fails as a strategy because the Rooted Colorful Motif problem has
no polynomial kernels on trees.
8

d

q
q
(n − k)
θ c
θ c
F

F P
{F}
{F}
X
Θ c
F

F
M θc
θ c
µ
θ c
{θ c }
{θ c }

F
F

G H H G
T 2 t K 1,t t = 7 θ c c = 7
a, b, c T
H
v ∈ S A u/∈ S A u/∈ S A
w
v u w R X
S A
k
S T

k
S T q q =(k + 1)
k = 2
v
(k + 2)
5 v
(k + 2) k = 3
v T ⋆ S
T
S v
t ⊕
t

f G
f G (S) S
G n H p
G n ⊕ H p
c 2c
2c
(s, p) b
v i
S ′ S = {x, y, z}
θ c
θ 9
β()
β()
θ c c = 7
β()
θ 5 k = 3
v k 2
H v
k O(1) v v/∈ H v
H v

c c = 5
c c = 5
v T ⋆
S T
T
F
{a, b, c, d, e, f, g}
{a, b, e, g}{b, e}{d, e, f}{b, g} {c, g}
B 1 B 2
A B A
B
ppt
θ c
c = 5 i
ppt
(x p ,y p )
1 p k

F i
C(F i )
Q
Q
R
T p ⊙ T q

F i
C(F i )
Q
Q
R
T p ⊙ T q

F i
C(F i )
Q
Q
R
T p ⊙ T q

F i
C(F i )
Q
Q
R
T p ⊙ T q

I
I
I ′ |I ′ | < |I|
P = NP
L ⊆ Σ ∗ × N Σ N
(x, k) L
(x, k) ∈ L f(k) · p(|x|) f k
p
L φ : Σ ∗ × N → Σ ∗ × N (x, k)
L (x ′ ,k ′ ) L

φ |x| k
|x ′ | |x|
L (x, k) ∈ L (x ′ ,k ′ ) ∈ L
(x, k)
(x ′ ,k ′ ) |x ′ | g(k) k ′ g(k)
g k
g(k)

Π
g
Π g(k)
g(k) =O(k 2 )
(g(k) =O(k))
g(k) =O(k O(1) )

H
H
H
H
F
H H ∈ F
F
G H H
G
H
H H
H
H H
H H ′
H ′ H

F
H H ∈ F
F
F
F F
F G F
F S ⊆ V(G) G \ S H
H ∈ F
F
F
G F
k
F S |S| k
F
F
F = {△}
F = {K 2,3 ,K 4 } F = {K 3,3 ,K 5 } F = {K 3 ,T 2 }
K i,j
i j K i

Deletion problem, Deletion we are asked problem, whether we are afamily subset askedF whether of ofat forbidden most a subset k minors. vertices of atWe can most obtain bekdelete
verti a n
from a graph G such from that a graph the resulting G such graph that Deletion the does resulting not problem contain graph when as does Fa minor contains not contain anya graph planar as a from min gra
the family F of forbidden the family minors. F of forbidden We obtainminors. a number We of obtain algorithmic a number results of algorithmic on the p-Fr
• alinearvertexkernelongraphsexcludingK
Deletion problemDeletion when F contains problema when planar F graph. contains We a planar give graph. We give
fixed integer.
• alinearvertexkernelongraphsexcludingK • alinearvertexkernelongraphsexcludingK • an approximation 1,t as an induced algorithm subgraph, 1,t as achieving an induced whereant subg app is
fixed integer. fixed integer. OPT is the size of an optimal solution on g
• an approximation
•
algorithm
an approximation
achieving
algorithm
an approximation
achieving
ratio
an approximation
of O(log 3/2 OPT),
ratio of
wher
O(l
OPT is the size of Finally, an optimal we obtain solution polynomial on general kernels undirected for the case graphw
OPT is the size of
an optimal solution
afixedintegerc.
on general undirected
Thegraphθ graphs.
Finally, we obtain polynomial kernels for the case when c consists of two ve
F contains the θ
Finally, we obtain polynomial kernels for the case when F contains the θ c graph as a minor c g
though this may appear to be a very restrictedfo
afixedintegerc. Thegraphθ c consists of two vertices connected by c pa
afixedintegerc. Thegraphθ c consists of two vertices connected by c parallel edges. Eve
though this may appear
well-studied
to be a very
problems
restricted
such
class
as Vertex
of problems
Cover,
it alr
though this may appear
well-studied
to be
problems
a very restricted Hitting such as
class
Vertex Set. ofThe problems
Cover, genericFeedback itkernelization already encompasse
Vertex algorithS
well-studied problems Hitting such as Set. Vertex The generic protrusion Cover, kernelization Feedback techniques, algorithm Vertex previously isSet based used and on only Diamond a non-tr for pr
Hitting Set. Theprotrusion generic kernelization techniques, previously algorithm T 2 is used based onlyon fora problems non-trivial onapplication topological og
protrusion techniques, Tpreviously 2
used only for K 1,7 problems on topological graph classes.
T 2
K 1,7
θ 7 .
K 1,7
T
θ 7 .
2 t K 1,t t = 7 θ c c = 7
θ 7 .
1 Introduction
1 Introduction
1 Introduction
i T 2 Let F be a finite set of graphs. In an p-F-Deletio
Let F be a finite set of graphs. In an p-F-Deletion problem 1 , we are g
G and an integer k as input, and asked whether at
Let F be a finite set G and of graphs. an integer In an k as p-F-Deletion input, and asked problem whether 1 , we atare most given k vertices an n-vertex can b
that the resulting graph does not contain a graph fro
G and an integer that k as the input, resulting and asked graphwhether does notat contain most ka vertices graph from canF be asdeleted a minor. from Mor
is defined as follows.
that the resulting is graph defined doesas not follows. contain a graph from F as a minor. More precisely the p
is defined as follows. F
p-F-Deletion
p-F-Deletion
Instance: AgraphG Instance: and a non-negative AgraphGinteger and a k. non-negative
p-F-Deletion
F
Instance:
AgraphG
Parameter:
k
and a non-negative
Parameter:
k
integer k.
Question: Does there Question: exist S ⊆Does V (G), there |S| ≤exist k, S ⊆ V (G), |S|
Parameter: k
such that G \ S contains such no that graph G \ from S contains F as ano mino gra
Question: Does there exist SF ⊆ V (G), |S| ≤ k,
Wesuch referthat
to such G \ Ssubset contains
S asno
F-hitting graph from
We refer to such subset set. FThe Sas asp-F-Deletion aF-hitting minor? set. problem The p-
F F
several fundamental several problems. fundamental For example, problems. when F For = {K example, 2 }, a complete when F
We refer to such subset this is Sthe asVertex F-hitting this Cover set. is The the problem. Vertex p-F-Deletion When Cover F problem = problem. {C 3 }, is a aWhen cycle generaliza onF thr =
several fundamental problems. For example, when F = {K 2 }, a complete graph on two ve
this is the Vertex ∗ Department of Informatics, University of Bergen, N-5020 Bergen, Norway. fomin
Cover problem. When ∗ Department F = of {CInformatics, 3 }, a cycleUniversity on threeof vertices, Bergen, N-502 this
† University of California, † University
San Diego,
of
La
California,
Jolla, CA
San
92093-0404,
Diego, La
USA,
Jolla,
dlokshtanov
CA 92093-
∗ Department of Informatics, ‡ The Institute of Mathematical Sciences, Chennai
F University of Bergen, ‡ The N-5020 Institute Bergen, Norway. of fomin@ii.uib.no.
Mathematical Scien
† {neeldhara|gphilip|saket}@imsc.res.in.
University of California, 1 San Diego, La Jolla, {neeldhara|gphilip|saket}@imsc.res.in.
CA 92093-0404, USA, dlokshtanov@cs.ucsd.edu
‡ We use prefix p to distinguish the parameterized version of the problem.
The Institute of Mathematical 1
We useSciences, prefix p to distinguish Chennai the parameterized - 600113, version
{neeldhara|gphilip|saket}@imsc.res.in.
F 1
O(log 1 We use 3/2 prefix OPT)p to distinguish the parameterized version of the problem.
1
t K 1,t
1
F
F
θ c c
F
F
F

F
F
2 O(k log k) n O(1) (2 kO(1) n O(1) )
F
F
F
F
F
F = {θ c }
F
θ c
θ c
k
k θ c
k
k
k

G
M S G
S
S M
H G H M
+
G
3 G
4 M |M|
M
G
M
G 3
|M|
O ∗ (2 |M| )

G =(V, E) k ∈ N c : V → [k]
k
G T k c T
k
F +
+
θ c
F

+
+

+
+

G H F H J
k l
V(G) G E(G)
M G G
G
u, v, x v G
v d(v) ∆(G)
G δ(G) G
n ∈ N [n] {1, . . . , n}
G (V, E) V E ⊆ V × V V
E

G ′ G V(G ′ ) ⊆ V(G) E(G ′ ) ⊆ E(G) G ′
G E(G ′ )={{u, v} ∈ E(G) | u, v ∈ V(G ′ )}
S ⊆ V(G) S G[S]
V(G)\S G\S N(S)
S V(G) \ S S
(u, v) G
u v
G H G
θ c
H G
φ : V(H) → 2 V(G) v ∈ V(H) G[φ(v)]
v, u ∈ V(H) φ(u) ∩ φ(v) =∅ (u, v) ∈ E(H)
(u ′ ,v ′ ) ∈ E(G) u ′ ∈ φ(u) v ′ ∈ φ(v)
G, H G ′ G
H G G ′ H G ′
H G G ′ H G
c ∈ N M G
θ c G M T 1 T 2 S c
T 1 T 2
C C C
C H H H/∈ C
k G
k G
G
γ(G) G
k
k 1
n n

V(G)
B G
(l × l)
⋆
S
⋆ B B
⋆ G G
G
(l × l)
T 1 ,T 2 ,...,T n T
V(T i ) ∩ V(T j ) ≠ ∅ ∀i, j.

a, b, c T
V(T 1 ) ∩ V(T 2 ) ∩ ...∩ V(T n ) ≠ ∅.
a, b, c T
⋂ s∈S V(T s) ≠ ∅.
S = {i : T i a, b, c}.
P 1 (a, b) T P 2 (b, c) T P 3
(a, c) T T V(P 1 ) ∩ V(P 2 ) ∩ V(P 3 )
x
T s s ∈ S P 1 ,P 2 P 3 x ∈ V(T s )
s ∈ S
n
n = 1, 2
n T n T 1 ,T 2 ,...,T n
T
a, b, c
a ∈ V(T 1 ) ∩ V(T 2 ) ∩ ···∩ V(Tn1),
b ∈ V(T 2 ) ∩ V(T 3 ) ∩ ···∩ V(T n )
c ∈ V(T 1 ) ∩ V(T n ).

T i a, b, c
V(T 1 ) ∩ V(T 2 ) ∩ ...∩ V(T n ) ≠ ∅.
L Σ ∗ × N Σ
(x, k) k
(x, k) f(k) · p(|x|) f
k p
Q ⊆ Σ ∗ × N (x, k) ∈ Σ ∗ × N
|x| + k (x ′ ,k ′ ) ∈ Σ ∗ × N (x, k) ∈ Q
(x ′ ,k ′ ) ∈ Q |x ′ | + k ′ g(k) g
g g
Q
G G (T, X = {X t } t∈V(T ) )
∪ t∈V(T ) X t = V(G)
{x, y} ∈ E(G) t ∈ V(T) {x, y} ⊆ X t
v ∈ V(G) T {t | v ∈ X t }

( max t∈V(T ) |X t | ) − 1 G
G (T, X)
T r X r = ∅
T
t t ′ X t ⊃ X t ′ |X t | =
|X t ′| + 1
t t ′ X t ⊂ X t ′ |X t | =
|X t ′| − 1
t t 1 t 2 X t = X t1 = X t2
t t X t = ∅
r T
G
O(|V(G)| + |E(G)|)
G t ∪ t ′X ′ t
t ′ t t H t G t [V(G t ) \ X t ]
v ∈ G T {t | v ∈ X t } T v
B ⊆ V(G) G T B := ∪{T v | v ∈
V(B)} T
T B
T
T T 1
T 2 i = 1, 2 B i ∪{v ∈ B | v ∈ X(T i )} X(T i )
T i B
G (u, v) u ∈ B 1 v ∈ B 2
X uv u v
T 1 T 2 T 1
T 2

G χ(G) (G) +1 χ(G)
G
|V(G)| |V(G)| = n n 2
χ(G) (G)+1 n
G |V(G)| = n+1 |V(G)| 2
v d(v) (G) T G
T ′ V(T) V(T ′ ) t
t ′ t
v ∈ X t \ X t ′ X t t
X t X t ′
t
T v X t
v
v d(v) (G)
v G d(v) (G).
χ(G \{v}) (G \{v})+1,
(G) +1 d(v) (G)
X ((G) +1) G \{v} v
X X
((G) +1) G \{v} v X
χ(G) (G)+1
G S ⊆ V(G) ∂ G (S) S
V(G) \ S S ⊆ V(G) S N G (S) =
∂ G (V(G) \ S)
+
r G X ⊆ V(G)
r G (G[X]) r |∂(X)| r

∨ ∧ ¬ ⇔ ⇒
∀ ∃
u ∈ U u U
d ∈ D d D
inc(d, u) d u
d u
adj(u, v) u v u u
v
p
p Π G k
S k
P Π (G, S)
φ
t
G t
S G φ f(t, |φ|)|V(G)|

∨ ∧ ¬ ⇔ ⇒
∀ ∃
u ∈ U u U
d ∈ D d D
inc(d, u) d u
d u
adj(u, v) u v u u
v
p
p Π G k
S k
P Π (G, S)
φ
t
G t
S G φ f(t, |φ|)|V(G)|

I NP
I ′ |I ′ | < |I|
NP
k
k
p(k) k
p(k) p(k) =O(k)
2k 67k
O(k 2 )
k
k
d
4k
2 k

φ
k
φ 3 n m
k m/2
k
m 2k n 6k
d
d
dd C d
U k
U ′ ⊆ U k U ′
C
k Y S 1 ,S 2 ···S k S i ∩ S j = Y
i ≠ j S i \ Y
F
U s |F| >s!(k − 1) s F k
F U

d
d O(k d d!d 2 ) (U, C,k)
d (U, C ′ ,k ′ ) |C ′ | O(k d d!d)
C S = {S 1 , ··· ,S k+1 }
k + 1 C k
Y S k
C ′ = C \ (S ∪ Y) (U, C,k) (U, C ′ ,k)
d ′ ∈ {1, ··· ,d}
k + 1
d ′ ∈ {1, ··· ,d} O(k d′ d ′ !) d
O(k d d!d)
G =(V, E) k
V ′ ⊆ V k (u, v) ∈ E
u ∈ V ′ v ∈ V ′
2 U = V C = {{u, v}|(uv) ∈
E} 4k 2 8k 2
4k
G =(V, E) V
C H R C H
C

C R N[C] ∩ R = ∅
E ′ C H E ′
|H| G ′ =(C ∪ H, E ′ )
H
G =(V, E) I ⊆ V |N(I)| <
|I| (C, H, R) G C ⊆ I
O(m + n) G I
(C, H, R) G =(V, E) G
k G ′ = G[R] k ′ = k−|H|
G V ′ k G
|H| C H H |V ′ ∩ (H ∪ C)| |H|
V ′ H ∪ C k − |H|
V ′′
k − |H| G ′ V ′′ ∪ H k G
4k
G =(V, E) k
M G V(M)
M |V(M)| > 2k
k |V − V(M)|
|V − V(M)| 2k 4k
|V − V(M)| >2k I = V − V(M)
|N(I)| |V(M)| < |I|
(C, H, R) G (C, H, R)

G ′ = G[R]
k ′ = k − |H|
|V − V(M)| 2k
4k
2k
2k
G =(V, E) k
k G E
n m G N(v)
v G N(v) :={u |{u, v} ∈ E} v
N[v] N(v) ∪ {v}
k
{u, v} C
u v C
k

{u, v}
N[u] =N[v] u
u
v
2 k
G =(V, E) 2 k
C 1 ,...,C k k v ∈ V b v
k i 1 i k v C i
2 k u ≠ v ∈ V b u = b v b u
b v u v
u v

{u, v}
N[u] =N[v] u
u
v
2 k
G =(V, E) 2 k
C 1 ,...,C k k v ∈ V b v
k i 1 i k v C i
2 k u ≠ v ∈ V b u = b v b u
b v u v
u v

q
q
q = 1
(A ⊎ B)
A S A
q q 1
q A
B
B q · |A|
q B
A
|B| q·m m
(n − k)

Q
H
M H
M
U X X R X
U X S A = X \ (U X ∪ R X ) U B
B T S A M
T = {x ∈ B |{u, x} ∈ M u ∈ S A }
S A |B| >mq U B B
H B
U B A U B
U X R X
M U B
S A S A

v ∈ A C(v) v v
C(v) ∩ S A = C(v) C(v) ∩ S A = ∅ v ∈ S A
v u U X {v, w} ∈ M v u
{u, w} ∈ E(H) v ∈ S A v ∈ S A
u R X {w, u} U X
u {w, v} ∈ E(H) U X
v v ∈ S A S = {v ∈ A|C(v) ⊆ S A }
G[S ∪ T] q S
H M v ∈ S
v ∈ S A u /∈ S A u /∈ S A
w v
u w R X
S A
T S G
T S A H
T S G T
S A H
v ∈ T u ∈ N(v) u /∈ S A u ∈ R X u ∈ R X
w
u v {u, v}
v ′ {v, v ′ } ∈ M v ′ ∈ S A
w ∈ U X v ′ u ∈ U X
w S
T
S A H

Q
q
(n − k)
q
S G \ S
k k
G
k
S ⊆ V(G) G \ S
|S| k
2k
3k q
q = 1
G
G M G M k
G k
A 2k
M B = V(G) \ A G[B]
B A H = (A ⊎ B, E ′ )

G A
q H q = 1
⎧⎪ ⎨
⎪ ⎩
k
k
M 1 H |M 1 | >k
|M 1 | k |B| |M 1 |
q q = 1
S ⊆ A, T ⊆ B H[S ∪ T] S
H G T S
G S T
S ∪ T G k := k − |S|
S ⊆ A, N(T) ⊆ S
T ⊆ B
S T

Q
|B| |M| = k G[A∪
B] |A ∪ B| 3k
3k
3k
(n − k)
G k (n − k)
G (n − k)
G (n − k)
χ(G) (n − k)
k
k k
k
(n − k)
G= (V, E)
k
V(G) (n − k)
3k−3
G
G G
(3k − 3) q
q = 1
v G G
G v l G
G G ′ G
G
G χ(G) (n − k)
χ(G ′ ) (n − k − l) G
G G

G M ′ G M ′ k
χ(G) (n − k)
M ′ G
|M ′ | n − 2|M ′ | G
n − 2|M ′ | G
n − 2|M ′ | + |M ′ | = n − |M ′ | =(n − k)
|M ′ | |M|
q S ⊆ A, T ⊆
B; S, T ≠ ∅ H[S ∪ T] S H
G T S χ(G) (n − k)
χ(G \ (S ∪ T)) (n − k − |T|)
G \ (S ∪ T) n ′ = n − |S| − |T| k
k − |S| χ(G) (n − k)
C : V → [n − k] G n − k G u T
B A \ S
S v ∈ S v C(u) u
v M u v G
u v G C(u) G
(n − k)
S T
χ(G\(S∪T)) (n−k−|T|)
S ∪ T G k := k − |S|
|B| |M| k−1 G[A∪B] |A∪B|
(3k − 3)
(n − k) (3k − 3)
k

Q
e =(u, v) ∈ E(G) e ′ =(u ′ ,v ′ ) ∈ E(G) {u, v}∩
{u ′ ,v ′ } ≠ φ G k
G S ⊆
E(G) G S
k
G= (V, E)
k
S ⊆ E(G) e ∈ E(G)
S |S| k
2k(k + 3)
q q (k + 1)
G
G M G
M
2k G
k A 4k
M B = V(G) \ A
G[B] B A
H =(A ⊎ B, E ′ ) G
A q H q =(k + 1)

⎫⎪ ⎬
⎪ ⎭
N(v)
|N(v)| >k
v
v
u v
k
M 1 H |M 1 | > 2k
|M 1 | 2k |B| > |M 1 |·(k+1)
q q =(k + 1)
S ⊆ A, T ⊆ B H[S ∪ T] |S| (k + 1)
S H G T S
u ∈ S k
u u
u
(k+1) u
k
u
u T
T S
T
(u, v u ) u ∈ S v u

Q
S ⊆ A, N(T) ⊆ S
T ⊆ B
S T q q = (k + 1)
k = 2
|B| < |M 1 | · (k + 1) =
(2k·k+1) G[A∪B] |A∪B| 2k(k+1)+4k =
2k(k + 3)
2k(k + 3)

Q
S ⊆ A, N(T) ⊆ S
T ⊆ B
S T q q = (k + 1)
k = 2
|B| < |M 1 | · (k + 1) =
(2k·k+1) G[A∪B] |A∪B| 2k(k+1)+4k =
2k(k + 3)
2k(k + 3)

k
F k θ 2
F {θ 2 }F
F k
G
k
S ⊆ V(G) G \ S G[S]
|S| k

K 4
W n n
n
v 1 ,...,v n
{v i ,v (i+1) } 1 i

(G, k)
H G
⋆
V(H) =V(G) ∪ {v e | e ∈ E(G)}
⋆
E(H) ={(x, v e ), (v e ,y) | e =(x, y) ∈ E(G)}
(H, k)
(G, k)
♦ S G S
H H G
H
♦ S H x ∈ S ⊆ V(H) v x ∈ V(G)
{ x x ∈ V(G)
v x =
z z ∈ N(x) x H
x/∈ V(G) N H (x)
G x
x
{v x | x ∈ S}
k G

⊲
(G, k)
(H, l) l
k H O(k 2 )
⊲ ∆
O(k∆)
(G, k)
v ∈ G
X
G (G \ X, k)
v
(k + 1)

G {x 1 ,y 1 ,z},...,{x k+1 ,y k+1 ,z}
(x i ,y i ) (x i ,z) (y i ,z) 1 i (k + 1)
△ i (x i ,y i ,z)
((G ∪△ 1 ∪ ...∪△ k+1 ),k+ 1) (G, k)
((G ∪△ 1 ∪ ... ∪△ k+1 ),k + 1) z
v (v, z)
v
v k
z
v
v
G x y z y ≠ x
z ≠ x y z x
y z y z
z x y
⇓
z y

(G, k)
u 1 ,u 2 ,...,u r
N(u 1 )=N(u 2 )=···= N(u r )={x, y},
r>(k + 2) H G
{u 1 ,u 2 ,...,u r } {u a ,u b }
(x, u a ), (x, u b ), (y, u a ), (y, u b ).
u a u b H (H, k)
u
1 u
2 ··· u
r
x
y
u
a
u
b
x
y
(k + 2)

H
⋆ V(H) =V(G)
⋆ (u, v) ∈ V(H) G x y
u v
H k G
k
k
G v ∈ V(G) l v
l G v
v
v
5 v
v (k + 1)
H v G v
(H, k)
(G, k)

→ G x x x
k (G \
{x},k− 1)
→ G x G
k
(G, k)
k = 0 G G
k
O(k 3 )
O(k 3 )
G
H
v 1 ,...,v r
(v i ,v i+1 ) i ∈ {1, 2, . . . , r − 1} v i G
x y v 1 v r
{v 1 ,...,v r } v x y
G {v 1 ,...,v r }
H
v i
x y v i

v H
x y
G
v i x y x y
v i 1 i r
H G u a u b
{u a ,u b ,x,y}
G x y
H
G k x y
S k k
{u 1 ,u 2 ,...,u r }.
r>(k + 2) i j G \ S u i u j
x − u i − y − u j − x
S x y
S H
S x y
{u 1 ,u 2 ,...,u r } x y
u i x y
u i
x y u i S S
x y
r H
r G r>k
k
(u, v) H G u
v G

H
r
H r G G
H
k
H k H
k + 2k(k),
k 2k
k
G
k + 2
G
(k + 2k(k))(k + 2) =O(k 3 ).
G m k
O(mk)
H (n−t)
t = O(k 3 ) r r>k
l
(G, k) (H, l)

(G, k)
G
v
v
v k
v ∈ V(G) H v ⊆ V(G) \{v}
v
v (k + 1) v
X ⊆ V(G) \{v} 2k v
H v v
q q =(k + 2)
(G, k) H v v
2k+(k+2)k 2k+(k+
2)k q q =(k + 2)
Greduction \ H v
v C 1 ,C 2 ,...,C r v
v H v
(G, k) k C i
C 1 ,C 2 ,...,C p
C p+1 ,...,C r
p k
p C i H v
u ∈ C i w ∈ H v (u, w) ∈ E(G)
H v
u C
H v G[C∪{v}] u ∈ C

C H v u
H v
v
H v
⋆
⋆
(k + 2) k = 3
G H v C C =
{c 1 ,...,c s } c i C i
s (r − p) (v, c i ) w ∈ C i
{v, w} ∈ E(G)
v h v H v G v
C i d(v) >h v +(k + 2)h v + p
(k + 2)h v
q q =(k + 2) A = H v B = C S ⊆ H v
T ⊆ D S |S| (k + 2) T N(T) =S
(v, u) u ∈ C i
c i ∈ T v w
w ∈ S v w
w ∈ S w v
w {w a ,w b }
(v, w a ), (v, w b ), (w, w a ) (w, w b ) w ∈ S
w a w b c i ∈ T v
C i

v
⋆
H v
⋆
v T ⋆ S
T
{w a ,w b } w ∈ S
v
q q =(k + 2)
G R
G S v G R
q (G, k)
(G R ,k)
G R Z
k Z G G
v ∈ Z S ⊆ Z
w ∈ S v/∈ Z w a /∈ Z w b /∈ Z
w a w b v/∈ Z w ∈ Z Z
{v, w a ,w b ,w} w ∈ S
v/∈ Z S ⊆ Z
v ∈ Z G R \{v} G \{v} Z \{v}
G R \{v} G \{v}
Z G

S v
Z v u
v G v G R
v ∈ Z Z v Z
k G S ⊆ Z
k G
k G R
G k G
k v S
v T
k
v S
W v
S x S W x 1 ,...,x k+2
x (k + 2) x
|W| k i, j x i /∈ W x j /∈ W
G\W {x, v, x i ,x j }
W
k

v
T W k
X ⊆ W W v
T v ∈ W X = ∅ W
v/∈ W S ⊆ W W\X
W \X k
W \X k |W| k
W\X W\X
X S
X {v} ∪ S
S v
2k+(k+2)k h v +(k+2)h v +p k+(k+2)k+k
h v k p
p k
∆
k∆ G (G, k)
G n (n − t)
δ ∆ G
n(δ−2)−tδ
2(∆−1)
F G E F
F G \ F n − |F| − 1
G \ F
∆|F| |E F | |E(G)| − n + |F| + 1>
(∆ − 1)|F| >
|F| >
(n − t) · (δ)
2
n(δ − 2) − tδ
,
2
n(δ − 2) − tδ
.
2(∆ − 1)
− n + |F|

v
T W k
X ⊆ W W v
T v ∈ W X = ∅ W
v/∈ W S ⊆ W W\X
W \X k
W \X k |W| k
W\X W\X
X S
X {v} ∪ S
S v
2k+(k+2)k h v +(k+2)h v +p k+(k+2)k+k
h v k p
p k
∆
k∆ G (G, k)
G n (n − t)
δ ∆ G
n(δ−2)−tδ
2(∆−1)
F G E F
F G \ F n − |F| − 1
G \ F
∆|F| |E F | |E(G)| − n + |F| + 1>
(∆ − 1)|F| >
|F| >
(n − t) · (δ)
2
n(δ − 2) − tδ
,
2
n(δ − 2) − tδ
.
2(∆ − 1)
− n + |F|

G ∆
c
G G ∈
G ∆ O(k∆)
G ∈ G H
(n − t)
H
δ
n(δ − 2) − tδ
,
2(∆ − 1)
(n − tδ)
2(∆ − 1) .
k

O(k 3 )
O(k 3 )

+
⊲
⊲
⊲
r G H
♦ H G \ H r
♦ H r

|V(H)|
H G \ H H
+
H
H
H
G \ H
X
G \ H H

H X
G (G \ H) ∪ ∗ X ∪ ∗
H
H
G =(V, E) S ⊆ V ∂ G (S) S
V\S S ⊆ V S N G (S) =∂ G (V\S)
r G =(V, E) X ⊆ V
r G |∂(X)| r (G[X]) r
r X X ′ = X \ ∂(X) r
X ′ X X X ′
t
t t G =(V, E)
t 1 t ∂(G)
G ∂(G)
G =(V, E) S ⊆ V
G[S] |∂(S)| ∂(S)
⊕ G 1 G 2 t
G 1 ⊕ G 2 t G 1 G 2
∂(G 1 ) ∂(G 2 )
G 1 ⊕ G 2
G 1 G 2

t ⊕

G G 1 G 2 t
G 1 ,G 2 ∈ G G 1 ⊕ G 2 G
G 1 ⊕ G 2 ∈ G G
G =(V, E) r
X X ′ X G 1 r
X ′ G 1 G G[V \ X ′ ] ⊕ G 1 G[X]
G 1 X ′ G 1
t
Π G t
G 1 G 2 G 1 ≡ Π G 2 c
t G 3 k G 1 ⊕ G 3 G 2 ⊕ G 3
(G 1 ⊕ G 3 ,k) ∈ Π (G 2 ⊕ G 3 ,k+ c) ∈ Π

Π
G t S t
S ⊆ G t G 1 G 2 ∈ S
G 2 ≡ Π G 1 S (Π,t)
t ≡ Π t
Π t ≡ Π
r q
r
r
+
H S
+
+
t
t G X

f G
f G (S) S
f G
: 2 X → N
f G (S) := max |I|.
I G
I∩X⊆S
c 1 c 2 X {c} c
f G
G S, S ′ ⊆ X |f G (S) − f G (S ′ )| t
S ⊆ X f G (∅) f G (S) f G (∅) +t

f G
T = {g | g : 2 X → {0, 1, . . . , t}} t G
X g ∈ T f G := 1 c + g c
G \ X t
G 1 G 2 ≡ T g ∈ T
f G1 := 1 c1 + g f G2 := 1 c2 + g c 1 c 2
G 1 \ X G 2 \ X
≡ T T
2 2t+1 ≡ T
Π t ≡ Π
Π =
≡ T ≡ Π G 1 ≡ T G 2
G 1 ≡ Π G 2
G 1 ≡ T G 2 G 1 ≡ Π G 2
t G 1 G 2 G 1 ≡ Π G 2
c t G 3 k (G 1 ⊕ G 3 ,k) ∈ Π
(G 2 ⊕ G 3 ,k+ c) ∈ Π
c 1 c 2 G 1 \ X G 2 \ X
c 1 c 2
c := c 2 −c 1 (G 1 ⊕G 3 ,k) ∈ Π G 1 ⊕G 3
k I 1 X X ∩ I 1
G 2 I 2 g(X∩I 1 )+c 2
X X ∩ I 1 I 2 ∪ (I 1 ∩ (V(G 3 ) \ X))
G 2 ⊕ G 3
|I 2 ∪ (I 1 ∩ (V(G 3 ) \ X))| g(X ∩ I 1 )+c 2 + k − (|I 1 ∩ V(G 1 )|
k + c 2 + g(X ∩ I 1 ) − (g(X ∩ I 1 )+c 1 )
= k + c 2 − c 1
= k + c
(G 2 ⊕ G 3 ,k+ c) ∈ Π
(G 2 ⊕ G 3 ,k + c) ∈ Π (G 1 ⊕ G 3 ,k) ∈ Π
≡ T ≡ Π

f G
T = {g | g : 2 X → {0, 1, . . . , t}} t G
X g ∈ T f G := 1 c + g c
G \ X t
G 1 G 2 ≡ T g ∈ T
f G1 := 1 c1 + g f G2 := 1 c2 + g c 1 c 2
G 1 \ X G 2 \ X
≡ T T
2 2t+1 ≡ T
Π t ≡ Π
Π =
≡ T ≡ Π G 1 ≡ T G 2
G 1 ≡ Π G 2
G 1 ≡ T G 2 G 1 ≡ Π G 2
t G 1 G 2 G 1 ≡ Π G 2
c t G 3 k (G 1 ⊕ G 3 ,k) ∈ Π
(G 2 ⊕ G 3 ,k+ c) ∈ Π
c 1 c 2 G 1 \ X G 2 \ X
c 1 c 2
c := c 2 −c 1 (G 1 ⊕G 3 ,k) ∈ Π G 1 ⊕G 3
k I 1 X X ∩ I 1
G 2 I 2 g(X∩I 1 )+c 2
X X ∩ I 1 I 2 ∪ (I 1 ∩ (V(G 3 ) \ X))
G 2 ⊕ G 3
|I 2 ∪ (I 1 ∩ (V(G 3 ) \ X))| g(X ∩ I 1 )+c 2 + k − (|I 1 ∩ V(G 1 )|
k + c 2 + g(X ∩ I 1 ) − (g(X ∩ I 1 )+c 1 )
= k + c 2 − c 1
= k + c
(G 2 ⊕ G 3 ,k+ c) ∈ Π
(G 2 ⊕ G 3 ,k + c) ∈ Π (G 1 ⊕ G 3 ,k) ∈ Π
≡ T ≡ Π

Π n 1 G n
V(G n )={x 1 ,x 2 }∪{a 1 ,...,a n }
E(G n )={(x 1 ,a 1 )}∪{(a i ,a i+1 ) | 1 i n−1} p 1 H p
V(H p )={y 1 ,y 2 } ∪ {b 1 ,...,b p }
E(H p )={(y 2 ,b 1 )} ∪ {(b i ,b i+1 ) | 1 i p − 1}
x 2
x 1
a 1 a 2 ··· a n
b p ··· b 2 b 1
y 2
y 1
G n H p
x 1 y 1
x 2 y 2 G n ⊕ H p
max{n, p}
y 2
b p ··· b 2 b 1 x 2 x 1 a 1 a 2 ··· a n
y 1
G n ⊕ H p
n

c k G
(G n ⊕ G, k) ∈ Π (G m ⊕ G, k + c) ∈ Π
c
c>0 k = m G = H m G n ⊕ G
m := max{n, m} (G n ⊕ G, k) ∈ Π
G m ⊕G m := max{m, m}
k + c (G m ⊕ G, k) /∈ Π
c

t G S ⊆ V(G)
(G ′ ,S ′ ) ∈ H t ζ Π G ((G′ ,S ′ )) P Π (G ⊕ G ′ ,S ∪ S ′ )
|S| ζ Π G ((G′ ,S ′ )) + f(t)
+ p
+ Π
c G k r
X G |X| >c O(|X|) G ∗ =(V ∗ ,E ∗ )
k ∗ |V(G ∗ )| < |V(G)| k ∗ k (G ∗ ,k ∗ ) ∈ Π
(G, k) ∈ Π
S (Π,2r) c = max Y∈S |Y| ζ
2r 2c S
2r H 2c H ≡ Π ζ(H) η
2r 2c N
2r H ′ k ′
(H ⊕ H ′ ,k+ η(H)) ∈ Π ⇐⇒ (ζ(H) ⊕ H ′ ,k) ∈ Π.
|X| >2c 2r X ′ ⊆ X c

X ′ b b
b cc H c
|V(G ∗ )| < |V(G)| H k ∗ (G ∗ ,k ∗ ) ∈ Π
(G, k) ∈ Π O(|X|)

X ′ b b
b cc H c
|V(G ∗ )| < |V(G)| H k ∗ (G ∗ ,k ∗ ) ∈ Π
(G, k) ∈ Π O(|X|)

F
F
r r
F

⎧
⎪⎨ V i V j
V 1 V 2 ··· V i ··· V p
|Vi| s
⎪⎩
U
V i U
⎧⎪ ⎨
⎪ ⎩
|N(V i ) ∩ U| (2b + 2)
(s, p) b
G b (s, p) G
G (p + 1) V 1 ,...,V p U
♦ p
1 r ≠ s p u ∈ V r v ∈ V s (u, v) /∈ E(G)
♦ p s |V i | s i 1 i p
♦ V i U (2b + 2)
i 1 i p
|N(V i ) ∩ U| 2b + 2,
V i U (2b + 2) G
d G d · (bsp)
(s, p)
G b
d G (d · bsp) s p G
(s, p)

G (d · bsp) (X,T)
G b G
b
T T ′ ⊆ V(T) X(T ′ ) ∪ q∈T ′X q
S ⊆ V T 2p 2p
T 1 ,...,T 2p T \ S i |X(T i )| 3s + b
T 1 ,...,T 2p
T \ S
T v ∈ T C T \{v} v
C v T
S T
r S = ∅ T r = T T r
T \S r i
v i V(T r ) T i
T r \{v i } v i |X(T i )| 3s + b v i S T r
v T r
v ∈ T r C T r \ v v
(3s + b)
2p
S T 1 ,...,T 2p
2p
T r = T
n
|X(T r )| n dbsp.
T v i V(T r )
T i T r \{v i } v i |X(T i )| 3s+b
T r v i T r x y
T r \{x} T r \{y} (3s + b)
v i T r

v i
y
x
⎨
⎩
(3s + b)
⎧⎪ ⎨
⎪ ⎩
2 · (3s + b)
⎧
v i
v i
y
x
{ }} { { }} {
((3s + b) · 2) + ((3s + b) · 2) +2,
x y
|X(T r )|
12s + 6b 12sb
2p
d 32 32 (12·2)+8
n>(12sb)(2p)+8(sbp)
(2p − 1)
n>(12sb) · 1 + 8(sb)p,

G
x
z
y
G
S ′ S = {x, y, z}
4(sb) (3s+b)
S T 1 ,...,T 2p S ′
S S ′ = S u v
S ′ t V T t S ′
l l
|S ′ | 2|S| v
v
S S
S
S |S|
S ′
|S ′ | 2|S| S ∗ = S ′ \ S
|S| >2p |S ′ | 2|S|
|S ∗ | 2p.
p T 1 ,...,T 2p
S ∗ p T 1 ,...,T 2p
S ∗ T 1 ,...,T p S ∗
i p T i S ∗ T ′
i = T i

T \ S ′ |X(T i ′)|
s T i v S ∗ v
3 |X(T i )| 3s + b T i \{v} T i ′ |X(T i ′ )| s
V i = X(T i ′)
U V i |V i | s
T i ′ S ′ u v
u = v T i ′ S ′ N(V i ) ⊆ X u ∪ X v
|N(V i )| 2b + 2
f
f : V(G) → N
ˆf : 2V → N
ˆf(S) = ∑ v∈S
f(v).
(f, s, p)
G b ˆf
(f, s, p) G G (p + 1)
V 1 ,...,V p U
♦ p
1 r ≠ s p u ∈ V r v ∈ V s (u, v) /∈ E(G)
♦ p ˆf(Vi ) s i 1 i p
♦ V i U (2b + 2)
i 1 i p
|N(V i ) ∩ U| 2b + 2,
G b
f : V(G) → N ˆf : 2V → N
ˆf(S) = ∑ v∈S
f(v).
d G (d·bsp)
s p G (f, s, p)

(2τ + 1) X τ
X G τ
X
X
G[X] X
⋆ X
⋆ ∂(X)
∂(X)
→ n G X (G[X]) τ
← 2(τ + 1) G
( )
|X|
.
4|∂(X)| + 1
G[X]
τ τ
G[X] (T, B = {B l } l∈V(T ) ) T
T
v ∈ ∂(X) l T v ∈ B l
|∂(X)| T
u v
T M T

G \ X
∂(X) ={x, y, z}
x
z
X
y
X
l l
|M| 2|∂(X)| v
v
∂(X)
|M| 2|∂(X)|
T T \ M 2|M| + 1
C 1 ,C 2 ...C η η
2|M| + 1
i C i M
M C i
T r i A i B i
r i A i
B i M B i A i
M
r i
T \ M A i B i
A i
C i
C i

• C i
T C i
C i
• C i
r i C i
r i
M C i
C i
C i
i η D i C i M
C i
D i := C i ∪ N T (C i ).
P i G D i
P i = ⋃
u∈D i
B u .
P i
P i ∂(X)
∂(X) C i N T (C i )
A B
A B v ∂(X) B v
A C i

C i A v
v
∂(X) C i N T (C i )
P i ∩ ∂(X) N T (C i )
(τ+1) C i 2(τ+1) ∂(X)
P i 2(τ + 1) G
η 2|M|+1 4|∂(X)|+1
|X|
P i M P 4|∂(X)|+1 1 ...P η
P i
θ c
θ c
F F
θ c
⋆ θ c
⋆ θ c
⋆
θ c
⋆ Θ c
T v k O(1)
v v/∈ T v
θ c
θ c
θ c
c ∈ N M G
θ c G M T 1 T 2 S c
T 1 T 2

θ c
θ 9
c 2 θ c M G
G M 2
M M
M
x M
x M T 1 S
x x T 1 θ c
M M
x M T 1 T 1 1 ,T2 1 ,...,Tl 1
T 1 x T 1 M ′
x M l>0 T i 1
T 2 l = 0 M ′ = T 2 M ′
x
φ
t
G t
S G φ f(t, |φ|)|V(G)|

θ c
θ c
θ c
θ c
M θ c
v v
H G φ H (G)
G H G
H V(H) ={h 1 ,...,h c } φ H (G)
φ H (G) ≡∃X 1 ,...,X c ⊆ V(G)[
∧
i≠j
(X i ∩ X j = ∅) ∧ ∧
∧
(h i ,h j )∈E(H)
1ic
Conn(G, X i )∧
∃x ∈ X i ∧ y ∈ X j [(x, y) ∈ E(G)]
]
conn(G, X) G[X]
G c
X 1 ,...,X c (h i ,h j ) ∈ E(H)
G X i X j
X 1 ,...X c x 1 ,...x c G[x 1 ,...,x c ] H
H G φ H (G)
G H φ H (G)
F F F
G
S ⊆ V(G)[ ∧ H∈F
¬φ H (G \ S)]

S G\S H
H ∈ F S
F
θ c
θ c
G v ∈ V(G) θ c l
v l θ c G v
v
v θ c
v M θ c
G v v
S ⊆ V(G) :
∃F ⊆ E(G)
⎡ ⎛
⎧⎪ ⎨ ⎢
⎣ ∀x ∈ S ⎜
⎝ ∃X ⊆ V ′ ⎪ ⎩
(G ′ ,X)
∧
x ∈ X
∧ ∀y ∈ S[y ≠ x =⇒ y/∈ X]
∧ φ θc (X ∪ {v})
⎫⎞⎤
⎪⎬
⎟⎥
⎠⎦
⎪⎭
S S
θ c
v F
G ′ V(G) F
V ′ = V(G) \{v} x ∈ S
X G x X ∪ {v}
θ c
S
G ′ x, y ∈ S X Y
X

Y u ∈ X u ∈ Y u
y Y G ′ y ∈ X X
G ′
S
F θ c
θ c
{θ c } F
G n v G
t ∈ O(1) G O(n)
S ⊆ V G G \ S θ c
{M 1 ,M 2 ,...,M l } θ c G
1 i

F
20 2l5
(l×l) S F G k (l×
l) H (G \ S) 20 2l5 (G)
k + d d = 20 2l5 (k + d)
S
G \ S
F = θ c
θ c
θ c
(2c − 1)
F θ c G F S
k (G) k +(2c − 1)
S F G k θ c ∈ F
(G \ S) (2c − 1) (G) k +(2c − 1)
(k + d)
S G\S

F
20 2l5
(l×l) S F G k (l×
l) H (G \ S) 20 2l5 (G)
k + d d = 20 2l5 (k + d)
S
G \ S
F = θ c
θ c
θ c
(2c − 1)
F θ c G F S
k (G) k +(2c − 1)
S F G k θ c ∈ F
(G \ S) (2c − 1) (G) k +(2c − 1)
(k + d)
S G\S

F P
F
O(OPT 2√ log OPT)
O(log 3/2 OPT)
{F}
F
H H ∈ F F
S G \ S
G F
S G \ S H
H ∈ F F
F =
{K 2,3 ,K 4 } F = {K 3,3 ,K 5 } F = {K 3 ,T 2 }

F P
K i,j i j
K i i T 2
F
F
F G F
F S ⊆ V(G) G \ S H
H ∈ F
F
F
G F
k
F S |S| k
F
F
F
F
F F
F
F
G F
k
F S |S| k

F
H F
h |H|
G H
H
S G H
G \ S
S
G H S
F
OPT
F
⊲ OPT
H
S
G H
⊲ (OPT · |S|)
H
H
H

F P
♦ H h (t × t) t =(14h −
24)
♦ 20 2t5 t×t
h H (G) >
20 2(14h−24)5 G H
F
d (20 2(14h−24)5 + 1)
d
G H G H
F
G H (d + 1)
F
G H
G G
F G
k
F h G
F S k (G \ S) d (G) k + d d =
20 2(14h−24)5
F
k
n
k

(G) d F G
d
l
(G) l d ′ (G) √ log (G),
d ′ l > (k + d)d ′√ log(k + d)
F G (k + 1)
G
(G) l (k + d)d ′√ log(k + d).
S G H
(d + 1)
(T, X = {X t } t∈V(T ) ) G
H t
t
t
H t := G t [V(G t ) \ X t ].
β : V(T) → N
β(t) =(H t ).
β
β
• t H t (H t )
• t s t H t H s
β
• t s t H t
H s H s H t

F P
t : {b, x, y}
{b, x, y}
{b, x, y}
{b, x}
{b, y}
{b}
{b}
{a, b}
β(t) =(G[{a, c}])
{a, b}
{b}
{a, b}
{a, b, c}
{a, b}
{a}
β()
• t r s H t
max{(H s ), (H r )},
r s β
β
β()
β

β(t) =(H t ) >d
t β(t) =(H t ) >d
P = V(H t ) Q =(V(G) \ P) \ X t S = X t X t
t G H G ∗
P Q G H
d H
(d + 1) t
β(t) >d β()
F G H
G H
G ∗ S
d

F P
(G, k)
(G) d
S F G
|S| >k
S
(T, X = {X t } t∈V(T ) ) l
l > (k + d) √ log(k + d) d
G F k
(T, X = {X t } t∈V(T ) )
V(G) G H G ∗ X t
T (G H )=(d + 1)
G H Y
Z (G ∗ ,k− (G H ))
Z |Y| >k
X ∪ Y ⋃ (G ∗ ,k− (G H ))

F
(G, k) F
O(k 2√ log k)
k G S
S F G
|S| = O(k 2√ log k)
S
l
(k + d) √ log(k + d) d
k
G k
G k
G ∗
G ∗ k G ∗
G G k
G F k
d
d

F P
(G, k)
α
(G) α · (k + d)
G H S
(G) > α · (k + d)
⋆ d

F
d (d + 1)
S(G H ) ∪ X t
⋃ ( (G ∗ ) ) ,
• G H
• S(G H ) F G H
• X t G H
• G ∗ (G \ G H ) \ X t
• |X t | (k + d)d ′√ log(k + d) X t
M H H G
• M H G H
• M H G ∗
• M H X t M H X t G H G ∗
M H
X t
M H
• S(G H ) G H
• (G ∗ ) G ∗ d
F G ∗
• X t
F G
O(k 2√ log k) S(G, k)
(G, k) G ∗ i G i H
G ∗ G H i th X i t
i th
S(G ∗ i,i)=OPT(G (i−1)
H
)+|S(G ∗ (i−1),i− 1)| + |X i t|.

F P
G ∗ 0 H
S(G ∗ 0 ,0)
S(G)
|S(G ∗ 0,0)| = 0.
OPT(G k H)+|S(G ∗ k,k− 1)| + |X k t |.
S(G) =(OPT(G k H)+OPT(G k−1
H
)+···+ OPT(G1 H)) + k · max{X i t}.
i
OPT(G i H ) >k 1 i k
OPT(G i H ) k Xi t = O(k √ log k) 1 i
k
S(G) =k 2 + k · O(k √ log k),
S(G) =O(k 2√ log k).
k
k G H G ∗ X t
k G H
G H
G H G ∗ X t
d

n
k 1 k n 1
k G F k − 1
OPT k S
O(k 2√ log k)
F OPT
F G
S ⊆ V(G) G[V \ S] F |S| =
O(OPT 2 · √OPT)
O(OPT ·√log
OPT)
O(log OPT 3/2 )
(G, k)
F
µ
⊲ µ k
⊲ X G A
G B µ(G A ) µ(G B )
µ(G)

F P
⊲ µ(G)
G A G B X
λ
λ · ,
k
log(k O(1) ) O(log k)
λ · O(log k).
O(k √ log k)
O(OPT · √log
OPT) O(log OPT 3/2 )
O(k 2√ log k)
m m =O(k 2√ log k)
µ(G) G
µ(G) := G.

G
(k + d) √ log(k + d)
t µ(H t ) H t
t t
µ H t
⊲ µ(H t ) t µ(H s ) s
t
⊲ µ(H t ) t µ(H s ) s
t v
H t H s
⊲ µ(H t ) t µ(H r )+µ(H s ) r s
t
µ(H t )
(m/α) t α µ(H t ) > (m/α)
µ(H s ) (m/α) s t t
t
µ()
t s H s G \ (H t ∪ X t )
X s µ(G \ H t ) m(1 −
1/α) m
H t (m/α) t µ(G \ (H t ∪
X t )) (m(1 − 1/α)) G \ (H t ∪ X t ) G \ H t
α 1

F P
t r s
t X t = X s = X r G A G B
X t
µ(H s ) m/α µ(H r ) m/α,
µ(H s )+µ(H r ) m
µ(H t ) > m/α,
µ(G \ (H t ∪ X t )) (1 − 1/α)m.
µ(H s ) µ(H r )
m/2α G A G B
µ(H s ) m/2α G A H s ∪ G \ (H t ∪ X t ) G B H r
µ(G B ) m/α
µ(G A ) m/2α +m(1 − 1/α) =(1/2α + 1 − 1/α)m =
(2α − 1)
2α
· m,
m α
µ(H s ) µ(H r ) m/2α
µ(H s ) > m/2α µ(H r ) > m/2α.
µ(H s )=m(1/2α + δ s ) µ(H r )=m(1/2α + δ r ).
µ(H s )+µ(H r ) m
(1/α + δ s + δ r 1) ⇒ δ s + δ r 1 − 1/α,

δ s δ r (1 − 1/α)/2
δ s α − 1
2α ,
H s ∪ G \ (H t ∪ X t ) H r
µ(G B ) m/α µ(H s ∪ G \ (H t ∪ X t ))
µ(H s ∪ G \ (H t ∪ X t )) m(1/2α +(α − 1)/2α)+m(1 − 1/α),
µ(H s ∪ G \ (H t ∪ X t )) m ·
(3α − 2)
.
2α
m (3α−2)

F P
d ′ l > (k + d)d ′√ log(k + d)
F G (k + 1)
G
(G) l (k + d)d ′√ log(k + d).
G A G B X
(T, X = {X t } t∈V(T ) ) G H t
t t
H t := G t [V(G t ) \ X t ].
β : V(T) → N
β(t) =|H t ∩ S|.
β
β
µ()
β
t
β(t) > (2/3)m G A G B G
• t s G A H s G B
G \ (H t ∪ X t )
• t s r
• β(s) m/2 G A H s ∪ G \ (H t ∪ X t ) G B H r
• β(r) m/2 G A H r ∪ G \ (H t ∪ X t ) G B H s
G A G B X
G A ∩Z G B ∩Z
k
k k

t : {b, x, y}
{b, x, y}
{b, x, y}
{b, x}
{b, y}
{b}
{b}
{a, b}
β(t) =G[{a, c}] ∩ Z
{a, b}
{b}
{a, b}
{a, b, c}
{a, b}
{a}
β()

F P
β
F
O(k(log k) 3/2 )
k F G
O(k(log k) 3/2 )
G j (1),G j (2),...,G j (2 j )
i
2 i
i
0 G G A
G B G
j
λ j (1), λ j (2),...,λ j (2 j )
2 j
λ j λ j
j
λ j = λ j (1)+λ j (2)+...+ λ j (2 j ).

k j (1),k j (2),...,k j (2 j )
F G j (1),G j (2),...,G j (2 j )
k F G
G
j (j + 1)
j
k j (1)+k j (2)+...+ k j (2 j ) k.
G j (i) 1 i 2 j
(k j (i)+d)
√
(k j (i)+d) log(k j (i)+d).
λ j (i)
√
λ j (i) =(k j (i)+d) log(k j (i)+d),
λ j
∑
√
(k j (i)+d) log(k j (i)+d).
1i2 j
(k + d(2 j )) √ log(k + d)

F P
k j (1)+k j (2)+...+ k j (2 j ) k
(k j (1)+d)+(k j (2)+d)+...+(k j (2 j )+d) k + d(2 j )
(
(kj (1)+d)+(k j (2)+d)+...+(k j (2 j )+d) ) √ log(k + d) (k + d(2 j )) √ (
log(k + d)
∑ ) √log(k
1i2
(k j j (i)+d) + d) (k + d(2 j )) √ log(k + d)
∑
(
∑1i2
(k j j (i)+d) √ )
log(k + d)
(k + d(2 j )) √ log(k + d)
1i2 j (
(kj (i)+d) √ log(k j (i)+d) ) (k + d(2 j )) √ log(k + d),
k j (i) k
λ j (k + d(2 j )) √ log(k + d).
j
(k + d(2 j )) √ log(k + d)
S(G, S) (G, S)
∑
1i2
|S(G j j (i),S j (i))| = ∑ 1i2
λ j j +
∑
|S((G
∑ 1i2j j (i)) A ,S j (i) ∩ (G j (i)) A )| +
1i2
|S((G j j (i)) B ,S j (i) ∩ (G j (i)) B )|
(G j (i)) A G j+1 (2i − 1) (G j (i)) B
G j+1 (2i)
(j + 1)
j S ∩ G A
|S(G j (i), ∅)| = 0
|S(G 1 (1), (S ∩ G 1 (1)))| + |S(G 1 (2), (S ∩ G 1 (2)))| + λ 0 ,

|S(G A , (S ∩ G A ))| + |S(G B , (S ∩ G B ))| + λ 0 .
γ
S(G) =λ 0 + λ 1 + ...+ λ γ .
λ j (k + d(2 j )) √ log(k + d),
(k √ log(k + d)) · γ + ∑
1jγ
d(2 j )
G A G B
|S ∩ G A | 5 6 |S|, |S ∩ G A| 1 3 |S|
|S ∩ G B | 2 3 |S|
O(log |S|) O(log(k 2√ k))
O(log k) γ
k
∑
1jγ
d(2 j )
(k √ log(k + d)) · γ

F P
(k √ log(k + d)) · (log k)
O(k(log k) 3/2 )
k 1 k n 1
k G F k−1
OPT k S
O(OPT(log OPT) 3/2 )
F OPT
F G
S ⊆ V(G) G[V \ S] F |S| =
O(OPT · (OPT) 3/2 )
(G, S)
S ∩ G = ∅
∅
(T, X = {X t } t∈V(T ) ) l
(T, X = {X t } t∈V(T ) )
V(G) G A G B X t
T |G A ∩ S| (5/6)|S| |G B | (2/3)|S|
(G A ,S∩ G A ) ∪ (G B ,S∩ G B ) ∪ X t
F
F
η
η η
G
η
1

η
O(log 3/2 n)

F P

{F}
F
F K 1,t
F F
K 1,t
t
K 1,t
(t−1) K 1,t K 1,3
K 1,7
K 1,t
O(k log k)
K 1,t
coNP ⊆ NP/poly
k
F h G F
S k (G \ S) d (G) k + d d =
20 2(14h−24)5

{F}
r
r F
r
+ Π
γ : N → N (G, k)
r X G γ(r) O(|X|)
(G ∗ ,k ∗ ) |V(G ∗ )| < |V(G)| k ∗ k (G ∗ ,k ∗ ) ∈ Π
(G, k) ∈ Π
G K 1,t
G ′ K 1,t
K 1,t
G
G G G ′
G
r
F
K 1,t
F

♦ X G τ τ
X (2τ + 2) X
X
(
)
|X|
.
4 · |∂(X)| + 1
X S
F S G
G \ S X
α
γ(α)
γ(α)
{(
}} ){
|X|
γ(2τ + 2)
4 · |∂(X)| + 1
|X| γ(2τ + 2)+4 · |∂(X)| + 1.
X ∂(X)
X
X G \ S (|X| + |S|) |S|
k
∂(X) S
♦ F h G K 1,t
S F ∂ G (G\S) ≡ N(S) g(h, t)·|S|
g h t

{F}
⎫
S
} F
|S| = (k)
X
(X) =d
⎪⎬
G
(|X|/q)
⎪⎭
(
γ(2d + 2) >
X
q
|X| qγ(2d + 2)
)
(q = 4N(S)+1)
N(S) |S| ·
(G ∗ ,k ∗ )
|V| = |X| + |S|
F
X
G K 1,t F
h S F X
G \ S
g h t
|∂(X)| g(h, t) · |S|,

X X S
S X
∂(X) = ⋃ v∈S
N X (v).
|∂(X)|
∑ v∈S |N X(v)|
|S| · max v∈S |N X (v)|.
v ∈ S h t
X = G \ S (X) d d = 20 2(14h−24)5
d d+1
|N X (v)| (t − 1)(d + 1)
G[N X (v)] (G \ S) d (G[N X (v)]) d
G[N X (v)] (d + 1)
κ G[N X (v)] d + 1
t
t K 1,t G
(d + 1) t
|N X (v)| (t − 1)(d + 1).
g(h, t) =(t − 1)(20 2(14h−24)5 + 1),
|∂(X) |S| · g(h, t)
F
+ p

{F}
F
F
F
p
F Π = F
t G ∂(G)
S ′′ ⊆ V(G) G G \ S ′′
F S = S ′′ ∪ ∂(G)
(G ′ ,S ′ ) ∈ H t ζ Π G ((G′ ,S ′ )) G ⊕
G ′ [(V(G) ∪ V(G ′ )) \ (S ∪ S ′ )] F |S|
ζ Π G ((G′ ,S ′ )) + t F
F
(G, k) F h
F
F O(log 3/2 OPT)
S G \ S F
S O(k log 3/2 k) (G, k)
F

X V(G) \ S d
G[X] d 20 2(14h−24)5
|X|
2(d + 1) Y G F
4|∂(X)|+1
γ : N → N
|X|
γ(2d+1) 2(d+1)
4|∂(X)|+1
Y G (G ∗ ,k ∗ ) |V(G ∗ )| < |V(G)| k ∗
k (G ∗ ,k ∗ ) F (G, k)
F G ∗ K 1,t
(G ∗ ,k ∗ ) X (2d+
2) γ(2d + 2) G ∗ \ X
O(k log 3/2 k)
k ∗ k
|V(G ∗ )| γ(2d + 2)(4|∂(X)| + 1)+|S|.
|∂(X)| g(h, t) · |S|
|X|
4|∂(X)|+1
|V(G ∗ )| = O(γ(2d + 2) · k log 3/2 k)=O(k log 3/2 k).
γ(2d + 2)
O(k log 3/2 k)
(2d + 2)
(2d + 2) Y
γ(2d + 2) ∂(Y) 2d + 2
k O(d) (G ∗ ,k ∗ )
G F X
k (G\X) d X
|V(G ∗ )| = O(k) G ∗
O(k)
G
K 1,t
F
O(k log k) K 1,t

{F}
G k
k G
l G l
S ⊆ V(G) (4l log l) G \ S
G k 2
S G \ S S
(8k log k) G k
+
G \ S (G ∗ ,k ∗ )
(G \ S) 1 (G, k)
V(G ∗ ) O(k log k)
O(k log k)
K 1,t

Θ c
θ c c 1
F θ c F
O(k 2 log 3/2 k)
F
Θ c
F
θ c ∈ F
θ c
S
G \ S
θ c
θ c c = 7
Θ c
c = 1
c = 2
c = 3
c = 2 O(k 2 )

Θ C
O(k 2−ε ) coNP ⊆ NP/poly
F
k
k O(1)
F
θ c
q
v ∈ V(G) T v
k O(1) θ c v v
θ c k
θ c v/∈ T v
G \{v}
θ c θ c G v
θ c
v
G \{v}
G n θ c v
G ′ = G \{v} θ c
v k T v O(k) v/∈ T v
G \ T v θ c T v

θ c (2c−1)
G ′ (2c − 1)
T v
β
l
(G) l d ′ (G) √ log (G),
d ′ (G ′ ) (2c − 1) =O(1)
(T, X = {X t } t∈V(T ) )
G H t
t
t
H t := G t [V(G t ) \ X t ].
M θc (G, v) θ c
G v M θc (G, v)
θ c v
β : V(T) → N
β(t) =M θc (H t ∪ {v},v),
β t θ c v
H t ∪ {v} G ′
v
β
β
• t H t (H t )

Θ C
• t s t H t H s
β
• t s t H t
H s θ c v (H s ∪
{v}) (H t ∪ {v})
• t r s
θ c H t
M θc (H s ∪ {v},v)+M θc (H r ∪ {v},v),
β()
β()
β() t θ c
v H t ∪ {v} β()
t
β() β()
t β(r) =0 t
G ∗ V(G ′ ) \ H t S
s s t
t s
G ∗ S
G ′ ∪ {v} θ c
θ c v
θ c v k
k
β()
θ c v

t : {b, x, y}
{b, x, y}
{b, x, y}
{b, x}
{b, y}
{b}
{b}
{a, b}
β(t) =M θc (G[{a, c} ∪ {v}])
{a, b}
{b}
{a, b}
{a, b, c}
{a, b}
{a}
β()

Θ C
θ c v G ∗
θ c v H t ∪ {v} S
(H t ∪ {v}) \ S θ c
H s ∪ {v}
θ c β(s) t
β()
k s 1 ,...,s k
1, . . . , k
G ′
G ′ G ′
|X si | = O(1),
X s s
k · (G ′ )=O(k)
k
(G, k) Θ c
(G ′ ,k ′ ) k ′ k
G ′ k
Θ c

v
θ 5 k = 3
v k 2
v θ c
(G, k)
G
G v ∈ V(G) θ c l
v l θ c G v
v
θ c k
v G v G
(G \{v}, (k − 1))
v
θ c
θ c
k
θ c k

Θ C
(G, k) Θ c (G, k)
G
k G θ c k θ c
O(k log 3/2 k) G
θ c k (G, k) Θ c
S O(k log 3/2 k)
H v θ c
v v/∈ H v
S θ c v ∈ S
S v := S \{v} G v := G \ S v
(2c−1) θ c v
G v θ c (G v ) (2c−1)+1 = O(1)
v G v
v ∈ S G v k+1
(G ← G \{v},k← k − 1)
(G, k)
v ∈ S
v G v k
v ∈ V(G) H v ⊆ V(G)\{v}
θ c v v
H v {v} v/∈ S H v = S
v ∈ S v G v
k T v O(k)
v θ c v G v
H v = S v ∪ T v |H v | h v H v
V(G) \{v}
v H v
q q = c
(G, k) S H v v
ch v +c(c−1)h v ch v +
c(c − 1)h v q q = c

Gv
⎧⎪ ⎨
⎪ ⎩
v
G \ S
(Gv) =O(1)
⎫⎪ ⎬
⎪ ⎭
G
Sv S \ v S θc
v
(G ∗ ,k ∗ )
k Gv \{v} θc
Hv ← Tv ∪ Sv
Tv
θc v Gv v/∈ Tv
Tv ∪ Sv
v G
Hv k O(1)
v v/∈ Hv Hv

Θ C
v H v θ c G H v
θ c G
v ch v + c(c − 1)h v
G \ H v v
C 1 ,C 2 ,...,C r v (c − 1)
θ c
H v θ c C i
θ c
C i H v u ∈ C i w ∈
H v (u, w) ∈ E(G)
H v G
θ c u C
H v G[V(C) ∪ {v}] θ c
u M ⊆ G v ∈ M
u ′ ∈ M u ′ /∈ C ∪ {v} v
u u ′ M c 2
θ c M G
D 1 ,D 2 ,...,D s i D i
H v
G
H v v ∈ V(G) w

v
H v
⋆
⋆
c c = 5
G H v D D =
{d 1 ,...,d s } d i D i
(v, d i ) w ∈ D i (v, w) ∈ E(G)
c v ch v H v v (c−1)
D i d(v) >ch v + c(c − 1)h v
|D| ch v q q =
c A = H v B = D S ⊆ H v T ⊆ D S |S| c
T N(T) =S
(v, u) u ∈ D i
d i ∈ T c v w w ∈ S
c v w c
v w c
v
ch v + c(c − 1)h v ,
h v := |H v | H v θ c θ c v
G H v D D = {d 1 ,...,d s }

Θ C
v
T ⊆ D
c c = 5
S ⊆ H v ,N(T) ⊆ S
v
⋆
H v
⋆
v T ⋆ S
T

v
⋆
H v
⋆
T
d i D i G \ H v x ∈ H v
d i ∈ D D i
x
q q = c A = H v B = D S ⊆ H v
T ⊆ D S |S| c T N(T) =S
♦ (v, u) u ∈ D i d i ∈ T
♦ v w w ∈ S
v w c v w (c − r)
r
q q = c G R
G S v G R
c (G, k) Θ c (G R ,k)
Θ c
G R Z k
Z θ c G v ∈ Z S ⊆ Z
v ∈ Z G R \{v} G \{v} Z \{v}

Θ C
G R \{v} G \{v} Z
k G S ⊆ Z
k G k
G R k
k v S
W v S
|S| c G[S ∪ T] v θ c
v G v
D D i
d i ∈ T X W
D i ∈ D W ′ W
X S W ′ := (W \ X) ∪ S
W ′ k S ′
S W W
W D θ c
c X ′ ⊆ X
S ′ X ′ |W ′ | |W| k
W ′ θ c G
M u ∈ X u ∈ D i i
M H v \ S v u H v \ S G \ S
M
q
S S G R
(G, k)
Θ c (G ′ ,k ′ ) k ′ k
G ′ O(k log 3/2 k)
(H, l)
(H, l)

(G, k)
(G ∗ ,k ∗ )
(G ∗ ,k ∗ ) ≡ (G, k)
(G ∗ ,k ∗ )
S O(k log 3/2 k)
v ∈ S v θ c k
(G \ S) ∪ {v}
v θ c k
(G ∗ ,k ∗ )
(G ∗ ,k ∗ )
T v
H v := T v ∪ S v S v := S \{v}
v/∈ S H v := S
[G, k, {H v | v ∈ V(G)]
(G, k) H
ch v + c(c − 1)h v O(k log 3/2 k)
v ∈ V(G) λ(v) v v c
q
λ(v) λ(v)
λ(v)
⎛
⎝ ∑
⎞
λ(v) ⎠ + n
v∈V(G)
∑
v∈V(G)
λ(v) n 2 ,

Θ C
(G, k)
X := (G, k)
G, k H v v ∈ V(G) X
v ∈ V(G)
d(v) >ch v + c(c − 1)h v
(G ∗ ,k ∗ )
(G ∗ ,k ∗ )
(G, k)
Θ c
O(k log 3/2 k)
(G, k) Θ c
(G ′ ,k ′ ) k ′ k
G ′ O(k log 3/2 k) θ c
S G ′ O(k log 3/2 k) d
S d := (G \ S)
θ c (2c − 1) d 2c − 1
2(d + 1) Y G ′ |V(G′ )|−|S|
4|N(S)|+1
Θ c γ : N → N
|V(G′ )|−|S|
γ(2d + 1)
4|N(S)|+1
2(d + 1) Y G ′ G ∗

|V(G ∗ )| < |V(G ′ )| k ∗ k ′ (G ∗ ,k ∗ ) Θ c
(G ′ ,k ′ ) Θ c
G
(G ∗ ,k ∗ ) S
(2d + 2) γ(2d + 2) G ∗ \ S
O(k 2 log 3 k)
|V(G∗ )|−|S|
γ(2d + 2) 4|N(S)|+1
k∗ k
|V(G ∗ )| γ(2d + 2)(4|N(S)| + 1)+|S|
γ(2d + 2)(4|S|∆(G ∗ )+1)+|S|
γ(2d + 2)(O(k log 3/2 k) × O(k log 3/2 k)+1)+O(k log 3/2 k)
O(k 2 log 3 k).
(G ∗ \
S) (2c − 1) =d G ∗ \ S d|V(G ∗ ) \
S| = O(k 2 log 3 k). S
|S| · ∆(G ∗ ) O(k 2 (log k) 3 )
O(k 2 log 3 k)
(2d + 2)
2d + 2 Y γ(2d + 2) ∂(Y)
2d + 2 k O(d) (G ∗ ,k ∗ )
∆(G ∗ )=O(k log 3/2 k) G θ c S
k (G \ S) (2c − 1) =d
S |V(G ∗ )| = O(k 2 log 3/2 k) |E(G ∗ )| O(k 2 log 3/2 k)
G ∗

Θ C
O(k 2 log 3/2 k) G
O(k 2 log 3/2 k)
2k O(k 2 )

F
F
F F
• S (k + 1) G
• G \ S G[S] F
• F k
+
G V(G) A B G[A] G[B]
S ⊆ B k G \ S
F
F

F
G \ S
S
G \ S G[S] F
F k G \ S
F
F
G F k F S
(k + 1)
k
F X |X| k S ∩ S = ∅
F
F O(k 3 )
F
F 2 O(k log k) n 2
F

F
G[B]
G \ B
F
F
F
F
H V(H) E(H)
V(H)
2 H I(H)
V(H) ∪ E(H) v ∈ V(H) e ∈ E(H) I(H)
v ∈ e

F
{a, b, c, d, e, f, g}
{a, b, e, g}{b, e}{d, e, f}{b, g} {c, g}
s h
K h s h
s h n K h
(s h
√
log h)n
s h = 0.319 . . . + o(1)
G K h
G 2 s h√
log h
n
H n
I(H) K h |E(H)| 2 s h√
log h
n
H
e H e
H e ′ ⊂ e H
H e e ′
I(H) e
H 2 H
G G V(H) G
H
G H G I(H)

G
I(H) 2 s h√
log h
n
H n
I(H) K h
H h 2 s h√
log h
h(h − 1)n/2
h h(h −
1)/2 I(H) K h
(s, p)
G b (s, p) G
G (p + 1) V 1 ,...,V p U
♦ p
1 r ≠ s p u ∈ V r v ∈ V s (u, v) /∈ E(G)
♦ p s |V i | s i 1 i p
♦ V i U (2b + 2)
i 1 i p
|N(V i ) ∩ U| 2b + 2,
G b d G
(d · bsp) s p G (s, p)
b d
G (d · bsp) s p G
(s, p) (s, p)

F
s p
s p
k
k O(1)
(G, S, k) F
c α β F
G (α(k + 1) +k) · c · (β(k + 1) +k)
r c r
F
G \ S G \ S
b (s, p)
G \ S s p
h
F F H
(V 1 ,...,V p ,U) (s, p) G \ S
α
p (s, p) (G\S) (α·(k+1)+k)
V i V i h
S
C 1 ,...,C γ G[V \ (U ∪ S)]
H S C i 1 i γ
e i = N(C i ) ∩ S
I(H) G \ U
C i
G \ U G \ U H I(H) H
G \ U H G
F S ∗ ⊆ (V \ S) k
G \ S ∗ H
H ∗ I(H ∗ ) H
H ∗ k H
I(H ∗ ) H ∗ (k + 1)

h H ∗
2 h√ s log h
h(h − 1)
· (k + 1).
2
α 2 s h√
log h
h(h − 1)/2
V i
h S
H h S
h H ∗
H H ∗
(α · (k + 1))
k
p (α·(k+1)+k)
h S
S
S
β s
(s, p) G \ S β · k
H ∗ H
(k + 1)
H
2 s h√
log h
· (k + 1).
H k H ∗
H
β 2 s h√
log h
2 s h√
log h
· (k + 1)+k.

F
e h H c[e]
C N(C) =e w(c[e])
C N(C) =e
β(k + 1)+k
β(k+1)+k
∑
i=1
w(c[e i ]) = s.
s>(β(k + 1) +k)c
c e
e
F
+
G 1 G 2 t
i ∈ {1, 2} f Gi V(G i )
[t] v ∈ G i f Gi
G 1 G 2 G 1
∼ =t G 2
h : V(G 1 ) → V(G 2 ) h (u, v) ∈ E(G 1 )
(h(u),h(v)) ∈ E(G 2 ) f G1 (v) =f G2 (h(v)) h
G 1 G 2
G t G t
1 t t G
µ G : V(G) → 2 [t] µ G
v l ∈ [t] {l} µ G (v) ={l}
v ∈ (V(G) \ ∂(G)) µ G (v) =∅

H
f H : V(H) → 2 [t] (u, v) ∈ E(H) H ′
H u v w uv
(u, v)
H H ′ f H ′ : V(H ′ ) →
2 [t] x ∈ V(H ′ ) ∩ V(H) f H ′(x) =f H (x) w uv
f H ′(w uv ) = f H (u) ∪ f H (v)
t
H f : V(H) → 2 [t] G
t µ G H
G H G
G
H t
G
H
h t
h t G M h (G) t
G h
t h G 1 G 2 t
G 1 ≡ h t G 2 M h (G 1 )=M h (G 2 )
t h t h
M h (G)
h t ≡ h t
t
≡ h t
h t h
h
h 2 (h 2) · (2 t ) h h
2 2(h 2)+th ≡ h t

F
F t G 1
G 2 G 1 ≡ F G 2 t G 3 Z ⊆ V(G 3 )
(G 1 ⊕ G 3 \ Z) F (G 2 ⊕ G 3 \ Z) F .
F q = max H∈F {|V(H)|}
t l =(q + t) ≡ l t ≡ F
G 1 ≡ l t G 2 G 1 ≡ F G 2 t ≡ F
G 1 ≡ l t G 2 G 1 ≢ F G 2
t G 3 Z ⊆ G 3
(G 1 ⊕ G 3 \ Z) F (G 2 ⊕ G 3 \ Z) F
(G 1 ⊕ G 3 \ Z) F (G 2 ⊕ G 3 \ Z) F
G 1 ,G 2 G 3
(G 2 ⊕ G 3 \ Z) F
H ∈ F (G 2 ⊕ G 3 \ Z) M (2,3)
H
φ : V(H) →
2 V(G′ )
v ∈ V(H) G ′ [φ(v)]
v, u ∈ V(H) φ(u)∩φ(v) =∅ uv ∈ E(H)
u ′ v ′ ∈ E(G ′ ) u ′ ∈ φ(u) v ′ ∈ φ(v) P 1 ,P 2 ,...,P h
M (2,3)
H Q i P i ∩ V(G 2 )
Q i P i ∂(G 2 )
G 2
t
∪ h i=1 Q i (h + t)
H ∗
∪ h i=1 Q i G 2 H ∗
l G 2 G 1 G 2
H ∗ l G 1 G 1
H ∗
R 1 ,...,R x X ∪ h i=1 Q i

∂(G 2 ) R j G 1
R 1 ,...,R x
M (1,3)
H
H G 1 ⊕ G 3 \ Z Y i P i ∩ V(G 3 )
M (1,3)
H
Y i R j
∂(G 3 ) {P i | P i ∩ ∂(G 3 ) ≠ ∅}
M (2,3)
H
X X B X ∂(G 3 )
X B R 1 ,...,R x
i X 1 ,...,iX m
∪ m j=1R i X
j
∩ ∂(G 3 )=X B .
M X ∪ m j=1 R i X ∩ V(G X
j
3 )
M X G 1 ⊕ G 3 \ Z
X B M X
M X \ X B X B
X G 2 ⊕G 3 \Z
(X ∩ V(G 2 ))
T 1 ,...,T m
{R i X
j
} m i=1 T i ∂(G 2 )
t i T i ∩ ∂(G 2 ) X
G 2 ⊕ G 3 \ Z P ij t i t j 1 i ≠ j m
P ij (X∩V(G 3 ))
T 1 ,...,T m T x G 2
T i T j G 2
(V(G 3 ) ∩ X) \ ∂(G 3 ) T x P ij
P ij
T x a b
a b R i X x
T x (G 1 ⊕G 3 \Z) {a, b} ⊆ R i X x
R i X x
a b
(G 1 ⊕ G 3 \ Z) Q ij t i
t j G 1 ⊕ G 3 \ Z M X
X M X M (1,3)
H
M(1,3)
H
H G 1 ⊕ G 3 \ Z

F
G 1 ⊕G 3 \Z F
≡ F ≡ h t
≡ F t
F c
G r X G |X| >c O(|X|)
G ∗ |V(G ∗ )| < |V(G)| Z ⊆ (V(G) \ X) ∪ ∂(X)
(G \ Z) F (G ∗ \ Z) F .
S ≡ F c =
max Y∈S |Y| ζ 2r 2c
S 2r H 2c H ≡ Π ζ(H)
|X| >2c 2r X ′ ⊆ X c

(G, Y, k)
P Π (G, T) T G k T ⊆ Y
F (G, G \ S, k)
G[S] G\S F
P Π (G, T) (∧ H∈F ¬φ H (G)) φ H (G)
H G
O(k 3 )
(G, S, k) F
p Y = G \ S
(G, S, k) G[S] G \ S
F Y (G ′ ,S,k)
Y ′ ⊆ G ′ \ S G F
Z ⊆ Y k G ′ F Z ′ ⊆ Y ′
k G ′ [S] G ′ \ S F
V(G)
(G, S, k) G[S] G \ S F Y
α β
F G
c·(α(k+1)+k)·(β(k+1)+k) r c
r
F
G c · (α(k + 1) +
k) · (β(k + 1) +k) c
X G
F p + 1
q O(|X|) Z ⊆ X ∩ Y
|Z| qk W ⊆ Y W (G, S, k)
W ′ W ′ X
Z W ′ ∩ X ⊆ Z (G, S, k)
Y (G, S, k) (Y \ X) ∪ Z
+ 2 Q X O(k)
O(k) X 1 ,...,X l X
∪ l i=1 X i = X Q

F
Q ∩ X i ⊆ ∂(X i ) Q Z ∪ (S ∩ X)
c qk + dl
X i X γ d
X γ ∩ Z ∂(X γ )
X γ
X γ \ ∂(X γ ) G
X γ F G X γ
G ′ G ′
(G ′ ,S,k) Y
G ′ \S F G\S
F O(n)
X γ
(G ′ ,S,k)
Y ′ ⊆ G ′ \ S |V(G)| = O(k 3 ) G ′ [S] G ′ \ S
F
O(k 3 )
O(k 3 )
O(n 2 )
F
p
F O(k 3 )
F

n G m
G F 11
+ G k G
F k F S l =
ηk log 3/2 k G F
k S v 1 ,...,v l V i =
{v 1 ,...,v i } ∪ (V(G) \ S) 1 i l G F Y
k Y ∩ V i F G[V i ] i l Y
F G[V i ] Y ∪ {v i+1 } F G[V i+1 ]
1 l i th G[V i ] S k + 1 G[V i ]
k
S
k Z G[S \ Z] F
(G, S \ Z, k − |Z|) F
F 2 O(k log k) n 2
2 k+1
F G[V i ] k
G F k F Y
k G[V i ] Y G[V i+1 ],Y∪ v i+1 ,k)
F 2 k+1
l
F
(G, S, k) F Y
G \ S
(G ′ ,S,k ′ ) Y ′
G F k G\S G ′
F k ′ k (G ′ \S)∩Y ′ |V(G ′ )| dk 3
k (G ′ \S)∩Y ′
( )
dk 3
k 2
O(k log k)
F
F O((2 k+1 2 k log k )2 k+1 l) S
O(n 2 )
11 +
O(2 O(k) n 2 ) F
2 O(k log k) n 2 11 +

F
O(n + m) =O(n 2 )
O(2 O(k) n log n) (G) >k
5k 11
+
S O(2 O(k) n 2 )

M θc
G H
♦ H G
H H
♦ H S ⊆ V(G) H
G \ S H
H
f : N → N k 0 G
H k H f (k)
H H = MH
H H = θ 1 H = θ 2 H = MH
♦ θ 1
k
2k

M θC
♦ H = Mθ 2
f(k) =O(k log k)
k
O(k log k)
H = Mθ c
c>0
Mθ c F F = θ c
θ c G S ⊆ V(G)
S θ c G \ S θ c
θ c
θ c G k
θ c θ c f(k) =O(k 2 log k)
θ c
Mθ c
G 2c 2 k 2 G k
θ c
G 2c 2 k 2 G k
θ c G θ c ηk 2 = O(k 2 )
η c
G k θ c
G 2c 2 k 2
G θ c f(k)

V(G)
B G
(l × l)
⋆
S
⋆ B B
⋆ G G
G
(l × l)
B
G (T, {T v : v ∈ V(T)}) G
B B B T B :=
∪{T v : v ∈ V(B)} T
{T B : B ∈ B}
T B i B j v
x ∈ T v ∈ T x G
x T Bi T Bj (u, v)

M θC
u ∈ B i v ∈ B j y ∈ T (u, v) ∈ T y
G
y T Bi T Bj
{T B : B ∈ B} T
x T x
V B B |T x |
G B
(1 + (G))
G k
(k + 1)
B G G
B
P G B
v P X
P v v P
P Z X Z
Q v X Z Q ∩ P = {v} P ∪ Q
Z P B
θ c
c
k
θ c

G 2c 2 k 2
G k θ c
G 2c 2 k 2 G
B 2c 2 k 2 + 1
P v 1 ,...,v t
P P t 2c 2 k 2 +1
P B B
1 i t B i B v i
1 i t ∪ i j=1 B j O i
s O s c 2 k 2 s
O 1 = 1 O t >2c 2 k 2 1 i t − 1 O i+1
O i + 1
B 1 = ∪ s i=1 B i B 2 = B \ B 1 O i
B 2 c 2 k 2
B 1 B 2 B
B P 1 P
v 1 v s P 2 v s+1 v t
P 1 P 2 c 2 k 2
P c 2 k 2
P 1 P 2 P 1 P 2
S c 2 k 2 S
B 1 B 2 c 2 k 2
A ∈ B 1 ,B ∈ B 2 A ∩ S = ∅ = B ∩ S A B
B A ∩ P 1 ≠ ∅,B∩ P 2 ≠ ∅ S P 1 P 2
P∪P 1 ∪P 2 k θ c E p
P 1 P 2 P i ∈
{1, 2} Q i = P i ∩ E p Q 1 Q 2 [M]
M = |Q 1 | = |Q 2 | f :[M] → [M] f(i) =j

M θC
v 1
v 2 ··· v s v s+1 ··· v t−1 v t
O 1 O 2 ··· O s O s+1 ··· O t−1 O t
|O
s | =(c 2 k 2 + 1)
v 1 ∈ B i1
v t ∈ B j1
v 1 ∈ B i2
v t ∈ B jt
v 1 ∈ B i3
···
···
v t ∈ B j3
v 1 ∈ B ip
v t ∈ B jp
B 1
B 1 B 2
B 2

P i j
C ⊆ P i, i ′ ∈ Q 1 ∩C; if(i ′ )
P f(1),f(2),...,f(M)
M
〈f(1),f(2),...,f(M)〉
√ |M| = ck i ∈ {1, 2} Q ′ i Q i
P
Q ′ 1 ,Q′ 2 P 1 ,P 2
k θ c
G 2c 2 k 2
k ′ = k − 1 θ c S ⊆ V(G) |S| =
O(k 2 ) G \ S θ c
G 2c 2 k 2 k ′ = k − 1
θ c O(k 2 )
µ
⊲ µ k
⊲ X G A
G B G A G B
⊲

M θC
G A G B
X X
G A G B
µ
O(k 2 )
µ
θ c G
µ
θ c
G
µ(G)
µ k O(1)
G (k − 1)
θ c
G
t µ
H t
t
t m µ(G)
µ H t
⊲ µ(H t ) t µ(H s ) s
t
⊲ µ(H t ) t µ(H s ) s
t v
H t H s

⊲ µ(H t ) t µ(H r )+µ(H s ) r s
t
µ(H t )
(m/3) t α µ(H t ) > (m/3) µ(H s )
(m/3) s t t
t
µ()
t s H s G \ (H t ∪ X t )
X s µ(G\H t ) m(1− 1 3 )
m H t
(m/3) t µ(G \ (H t ∪ X t )) (m(1 − 1 3 ))
G \ (H t ∪ X t ) G \ H t
t r s t
X t = X s = X r G A G B
X t
µ(H s ) m/3 µ(H r ) m/3,
µ(H s )+µ(H r ) m
µ(H t ) > m/3,
µ(G \ (H t ∪ X t ))
(
1 − 1 )
m.
3
µ(H s ∪ H r ) (2/3)m µ(G \ (H t ∪ X t )) (2/3)m
G A := (H s ∪ H r ) G B :=
G \ (H t ∪ X t )
G S G \ S
G A G B
µ(G A ) (2/3)µ(G) µ(G B ) (2/3)µ(G).

M θC
2c 2 k 2
(k/2) 2c 2 (k/2) 2
µ
θ c d
20 2t5
t×t (d+1)c×d
d θ c 20 2t5
t = dc
θ c d
θ c d
20 (2dc)5
µ(G) =1
t
0 1 X t
t θ c
G\X H t H 1 ,...,H p H p+1 ,...,H q
G \ X θ c H i 1 i p
t µ() 0 1
θ c H i p

µ(G) =1 G θ c
θ c
θ c
(G, µ(G))
(T, X = {X t } t∈V(T ) ) G
(T, X = {X t } t∈V(T ) )
µ(G) =1
V(G) G A G B X t
T µ(G A ) (2/3)µ(G) µ(G B ) (2/3)µ(G)
(G A , (2/3)µ(G)) ∪ (G B , (2/3)µ(G)) ∪ X t
O(k 2 )
θ c
t G
X t
θ c G A G B
S 1 S 2
G \{S 1 ∪ S 2 }
G A G B X t
X t G A G B S 1 ∪ S 2 ∪ X t
θ c G

M θC
S(G, µ(G)) S(G A , (2/3)µ(G)) + S(G B , (2/3)µ(G)) + (G).
(G) c 2 µ(G) 2 µ(G)
k k µ(G)
S(G, µ(G)) T(G, k)
T(G, k) =T(G A ,2/3k)+T(G B ,2/3k)+c 2 k 2 .
k = 1
T(G, 1) =O(1).
T(G, 1) =O(1)
T(G, k) =c 2 k 2 +
( d∑
i=1
( 8
9) i
· c 2 k 2 )
+ O(1)
i → ∞
d∑
) i
.
i=1
( 8
9
O(k 2 )
{θ c }
k k
θ c G

{θ C }
θ c
G k
k
S 1 ,S 2 ,...,S r ⊆ V r k G[S i ] θ c
E(S i ) ∩ E(S j )=∅ 1 i r
θ c
v H v v
v
v
θ c
G k θ c θ c
f(k) =O(k 2 )
θ c
θ c O(k 2 ) θ c O(k 2 log k)
θ c
θ c O(k 2 log k)
S
S
v
θ c G v
r r v r
V(G)
{S 1 ,S 2 ,...,S r }

M θC
v
S i ∩ S j = {v}, ∀1 i, j r
G[S i ] θ c 1 i r
k v G
r
r r
θ c k
k θ c
v
θ c
θ c k
θ c
k
v ∈ S S v := S \{v} G v := G \ S v
(2c − 1) θ c
v G v θ c c (G v )
(2c − 1) +1 = O(1)
v G v v ∈ S G v
k + 1
v ∈ S v G v
k
v ∈ V(G) H v ⊆ V(G)\{v}
θ c v v H v
{v} v /∈ S H v = S
v ∈ S v G v
k T v O(k)
v θ c v G v
H v = S v ∪ T v |H v | h v H v

{θ C }
V(G) \{v}
v H v
v H v θ c G H v
θ c G G \ H v
v C 1 ,C 2 ,...,C r v
(c − 1)
θ c H v θ c
C i θ c
C i H v u ∈ C i w ∈
H v (u, w) ∈ E(G)
H v G
θ c u C
H v G[V(C) ∪ {v}] θ c
u M ⊆ G v ∈ M
u ′ ∈ M u ′ /∈ C ∪ {v} v
u u ′ M c 2
θ c M G
D 1 ,D 2 ,...,D s i D i
H v
G
H v v ∈ V(G) w
G H v D D =
{d 1 ,...,d s } d i D i
(v, d i ) w ∈ D i (v, w) ∈ E(G)
k
θ c v v
H v
{u 1 ,...,u hv }
H v G

M θC
u i x i
u i u i
u j j

{θ C }
k O(1)
θ c

M θC

⊆ /
Q, Q ′ ⊆ {0, 1} ∗ Q Q ′
D x =(x 1 ,...,x t )
x i ∈ {0, 1} ∗ i ∈ [t] D(x) ∈ {0, 1} ∗
|D(x)| =(max i∈[t] |x i |) O(1)
f
f(k)p(n)
k

D(x) ∈ Q ′ i ∈ [t] :x i ∈ Q
Q ′ = Q Q
Q Q ′ Q ′
NP
⊆ /
Π
Π A x =
((x 1 ,k), (x 2 ,k),...,(x t ,k)) x i ∈ {0, 1} ∗ i ∈ [t]
A(x) :=(y, l) ∈ {0, 1} ∗ × N
l = k O(1)
(y, l) ∈ Π i ∈ [t] :(x i ,k) ∈ Π
NP
⊆ /
Π
P NP Π
P

k
G k
k
G k
2 O(k) n O(1)
k
c : V → [k]
k
k
k
(G 1 ,k),...,(G t ,k)
G ′
G ′ k i 1
i t G i k
k k
k
⊆ /

P Q
P Q P ppt Q
f : {0, 1} ∗ −→ {0, 1} ∗
p : N → N x ∈ {0, 1} ∗ k ∈ N f(x, k) =(y, l)
(x, k) ∈ P (y, l) ∈ Q
l p(k)
f P Q
f(k)p(|x|) (x, k) f
p
|x| k
(A, k) (B, l) A
NP B ∈ NP
A B B A
(A, k) (B, l)
A NP B ∈ NP A ppt B A
B ⊆ /
A
NP B A

x
A
z ∈ A
NP
f(x) =y
B
K(y)
A B A B
G =(V, E) k
k
G k
G =(V, E) k
k
G k

k
k
k k
k
O(k log k)
O(k 2 )
O(k 3 )
k
L k {1, 2, . . . , k} L ∗ k
L k x ∈ L k x w 1 ···w r ∈ L ∗ k w i ···w j ; 1
i

(W, k) W = w 1 ···w n {0, 1} ∗
G =(V, E) n v 1 ,...,v n
{v i ,v i+1 } 1 i

+ k
n O(k 3 )
t
t Π
(I, k) |I| t k t (I, k) ∈ Π
Π g(k)
(I, k) g(k) Π
(I, k) ∈ Π |I| k |x ′ |,k ′

θ c
⊆ /
θ c
G k
G k
k
⊆ /
c θ c
c c
θ c
θ c
G k
k
G k H 1 ,H 2 ,...,H k
1 i k H i θ c
H i θ c

c = 1 θ c
k G
G c =
2
⊆ /
c 3
⊆ /
G S
G \ S
♦ k
♦ k
+
{θ c }
θ c

{θ C }
c 3 θ c
k d d
⊆ /
θ c
⊆ /
k L k
{1, 2, . . . , k} L ∗ k L k x ∈ L k
x w 1 ···w r ∈ L ∗ k w i ···w j ; 1 i

θ c
c = 5 i
v j v j ; 1 j n S(v j ) (c − 2)
v j D(v j )
(G, k) G k
θ c
(w = x 1 x 2 ...x n ,k)
(G, k) θ c (w, k)
(w, k)
(G, k) θ c
l = x i , ···x j = l l w 1 l k
H G {D l ,v i ,...,v j } ∪ S(v i )
θ c c
D l v i H (D l ,v i ) (c − 2)
〈D l ,y,v i 〉 ; y ∈ S(v i )
(D l ,v j ) v j v i G
θ c l
x i ...x j w
l v i w

{θ C }
G k
θ c
G k θ c
D i ; 1 i k G
θ c G
k
G
l; 1 l k M l θ c G D l
P l = v i ···v j L k (v i )=L k (v j )=l
x i ···x j l w M l ; 1 l k P l
M l w
M l ; 1 l k P l
1 l k
M l v i ,v j
L k (v i )=l = L k (v j )M l M l D l
D l
M l
M l v i
M l 2 D l
M l M l
v i M l v i
M l G D l v i S(v i )
θ c M l
v i G

w L k (w) ≠ l w
2 M i S(w) M i
M i M i w v
G
M i M l v i ,v j
L k (v i )=L k (v j )=l M l
v i v j M l G D l v i ,v j
S(v i ) ∪ S(v j )
M l D l
M l
v i v j

⊆
/
(W, k) W = w 1 ···w n {0, 1} ∗
G =(V, E) n v 1 ,...,v n P
G i ∈ L k x i x i
v j w j = i i
n u 1 ,...,u n i ∈ L k
y i y i u j w j = i
{u i ,u i+1 } 1 i

w = 1122312 ←
y 2
1 1 2 2 3 1 2
u 1 s 1 u 2 s 2 u 3 s 3 u 4 s 4 u 5 s 5 u 6 s 6 u 7
1 1 2 2 3 1 2
v 1 r 1 v 2 r 2 v 3 r 3 v 4 r 4 v 5 r 5 v 6 r 6 v 7
ppt
x 2
w k L G k
w k L G k
i ∈ L i
G i w p ···w q
x i v p s p u p+1 ···s q u q y i x i .
Q
(x i ,v p ) (v p ,s p ) (u q ,y i ) (y i ,x i )
Q
Q
i j w i = w j

Q
(u p ,s p )
x i y i v p
w p i v p
w
Q w p+1 ...w q
w s p
s p P
s p s p v p P
v p s p
G k
w k L
G k w
k L
C 1 ,...,C k k
C j x p y p 1 p k
G {(x p ,y p ) | 1 p k}
V(G) x p
y p P Q
G
E C := {(x p ,y p ) | 1 p k}.
C j E C
k E C C 1 ,...,C k

C i (x p ,y p )
C j x p y p 1 p k
{x p ,y p } Q p
w = 1122312
y 1
y 2
1 1 2 2 3 1 2
u 1 s 1 u 2 s 2 u 3 s 3 u 4 s 4 u 5 s 5 u 6 s 6 u 7
1 1 2 2 3 1 2
v 1 r 1 v 2 r 2 v 3 r 3 v 4 r 4 v 5 r 5 v 6 r 6 v 7
x 1
x 2
(x p ,y p )
1 p k

G
M S G
S
S M H G
H M
+
G
3 G 4
M |M|
M
G
M
G 3
|M|
O ∗ (2 |M| )
3 3

⊆ /
k
+
⊆ /

⊆ /
k
+
⊆ /

⊆ /
k
+
⊆ /

U F = {F 1 ,F 2 ,...,F n } U
i, j F i ∪ F j = U
C : F → U C(F i ) ∈ F i
R ⊆ F ⋃ S∈R
S = U C R
U F = {F 1 ,F 2 ,...,F n } U
i, j F i ∪ F j = U

R ⊆ F ⋃ S∈R S = U
U F = {F 1 ,F 2 ,...,F n } U
k
R ⊆ F |R| k
S∈RS ⋃ = U
k = 1 i F i = U
k>1 ( n
2)
i, j F i ∪ F j = U
(U, F =
{F 1 ,...,F n },k)
(U ′ , F ′ ,C)
F ′ k F
X = {x 1 ,x 2 ,...,x k } U 1 i n, 1 j k
F ij = F i ∪{x j } 1 i n, 1 j k C(F ij )=x j U ′ = U∪X
F ′ = {F ij | 1 i n, 1 j k} U ′ = U ∪ X
i, j, p, q F ij ∪ F pq = U ′
R = {F i1 ,F i2 ,...,F it } k
R ′ = {F i1
∪ {x 1 },F i2 ∪
{x 2 },...,F it
∪ {x t }} 1 j k F ′ j = {F ij | 1 i n} t < k,
F ′ j ,t < j k R′ .

R ′′ . R ′′
(U ′ , F ′ , C) d ∈ X F ′ j
d R ′′ d ∈ U. d ∈ F ij ∈ R
F ij R F ij ∪ {x j } ∈
R ′ ⊆ R ′′ d R ′′
R ′′ R ′′
F ′ j R ′′ = {F ′′
1 ,F′′ 2 ,...,F′′ k} R = {F ′′
1\X, F ′′
2\X, . . . , F ′′ k\X}
d ∈ U F ′′
i ∈ R′′ F ′′
i \X ∈ R |R| k,
(U, F = {F 1 ,...,F n },C)
⋃
S∈F S = U T F i ∈ F
x ∈ F i x[i] V(T) U
F i ,1 i n
F i u i x[i]

F i r r
u i T
V(T) ={x[i] | x ∈ F i ,1 i n} ∪ {u 1 ,...,u n } ∪ {r}
E(T) ={(x[i],u i ) | x ∈ F i ,1 i n} ∪ {(u i ,r) | 1 i n}
F i
C(F i )
x ∈ U c x
f i C(F i ) c
x[i],x∈ F i c x u i c fi F i
u i r c r
c(x[i]) = {c x | x ∈ F i ,1 i n}
c(u i )={c fi | 1 i n c(r) =c r }.
U
c r T (T, |U| + 1, c)
R = {F i1 ,F i2 ,...,F it } ⊆ F T ′
T u i1 ,...,u it r
(r, u i1 ),...,(r, u it ) u j T ′

u j
T ′′ .
T ′′
c x x ∈ U ∪ {r} T ′′ c x
c x = c r T ′′ r c x ≠ c r . F p
R x ∈ U
C(F p )=x
T ′′ u p c x C(F p ) ≠ x
T ′′ T ′′ u p
c x T ′′ T |U| + 1
T ′′ T |U| + 1 {u i1 ,u i2 ,...,u it } =
V(T ′′ ) ∩ {u 1 ,...,u n } R = {F i1 ,F i2 ,...,F it }.
R {c(u i1 ),c(u i2 ),...,c(u it )}
d ∈ U T ′′ |U| + 1
T ′′ c d c d ≠ c r . T ′′ c d
u i u i . F i R
R d
(T, C) T
u 1 ,...u r T V i
u i U i V i ∪ {u i } X ⊆ V(T) c(X)
X
c(X) ={d | ∃x ∈ X, c(x) =d}.
i ≠ j
(c(U i ) ∪ c(U j )) \ C ≠ φ.
i, j F i ∪ F j = U

k
k
(T, k, c)
r r T
r
T(i) i th r T
T(i) :=T[v i ∪ ( N[v i ] \ r ) ].
T(i) i th
T
(Q, k, c q )
(T, k, c) Q Q T
( k
2)
V(Q) =V(T) ∪ {v[i, j] | i, j ∈ [k] i ≠ j}.
X {v[i, j] | i, j ∈
[k] i ≠ j} r c q (u) =c(r)
u ∈ V(Q) \ V(T) u V(T) c q (u) =c(u)

Q T
T(i) T(j) v[i, j] T(i) ∪ T(j)
v[i, j]
r v[i, j]
∀{u, v},u ∈ T(i),v ∈ T(j) i, j ∈ [k] i ≠ j (u, v) ∈ E(Q)
(u, v) ∈ E(T)
∀i, j ∈ [k],i≠ k u ∈ T(i) ∪ T(j) (v[i, j],u) ∈ E
∀u ∈ X ∀v ∈ N[r] (u, v) ∈ E
∀u, v ∈ X (u, v) ∈ E
Q
Q
T(i) T(j) i ≠ k + 1
j ≠ k + 1 v[i, j] (u, v)
u ∈ T(i) v ∈ T(j) v[i, j]
T(i) T(j)
T(l) v[l, u] u
T(l) v[i, j] v[i, j]
v[l, u] T(l)
S
v S v

(T, k, c)
(Q, c q ,k)
T ′ T Q R
Q R
R
Q R \ v[i, j] T
T
R T ′ T
k
(k + 1)
(T, c, k) k T(i)
T i
T(i) ={v ∈ T | c(v) =i}.
T(i) i
(R, c r ,k+ 1) T R
(Q, c q ,k+ 1) (T, c, k)
Q T ( k
2)
V(Q) =V(T) ∪ {v[i, j] | i, j ∈ [k] i ≠ j}.
(k+1) c q (u) =k+1 u ∈
V(Q) \ V(T) T(i) i Q i ∈
[k + 1] u V(T) c q (u) =c(u) Q
T
T(i) T(j)
v[i, j] T(i) ∪ T(j)

i ∈ [k + 1] ∀{u, v} ∈ T(i) (u, v) ∈ E(Q)
∀{u, v},u ∈ T(i),v ∈ T(j) i, j ∈ [k] i ≠ j (u, v) ∈ E(Q)
(u, v) ∈ E(T)
∀i, j ∈ [k],i≠ k u ∈ T(i) ∪ T(j) (v[i, j],u) ∈ E
Q
Q
T(i) T(i)
T(j) i ≠ k + 1 j ≠ k + 1
v[i, j] (u, v) u ∈ T(i)
v ∈ T(j) v[i, j]
T(i) T(j) T(l)
v[l, l ′ ] l ′ ≠ l T(l) v[i, j]
T(k + 1) v[l, l ′ ] T(l)
(R, c r ,k+ 1) R
Q v[i, j] ∈ T(k + 1)
V[i, j] ={v[i, j] (1) ,v[i, j] (2) ,...,v[i, j] (d ij) },
d ij v[i, j] V(R) \ T(k + 1)
N(v[i, j])
(v[i, j] (l) ,u l ) u l l th v[i, j]
T(k + 1) u V(Q) c r (u) =
c q (u) u ∈ T(k + 1) c r (u) =k + 1

R
R
T(i)
T(i) T(j) i ≠ k + 1 j ≠ k + 1
u ∈ T(i) l th
u
v[i, j] Q v ∈ T(j) l th
v
v[i, j] (u, v[i, j] l u
) ∈ E (v[i, j] l u
,v[i, j] l v
) ∈ E (v[i, j] l v
,v) ∈
E u v
v[i, j] l T(i) T(j) T(l)
(T, c, k)
(R, c r ,k + 1)
T ′ T R ′ R
R ′ T ′ ∪ {(u, v)} u ∈ T ′ v ∈ T(k + 1) (u, v) ∈ E
v R ′ R u
R ′ T(k + 1) u R ′
u ∈ T(k + 1) T(k + 1)
R ′ \{u} T ′ T
⊆ /
⊆ /

G =(V, E)
(T, k, c)
T
c : V(T) → [k]

T p T q l p ∈ V(T p )
T p f q ∈ V(T q ) T q T p ⊙ T q
V(T p ⊙ T q )=V(T p ) ⊎ V(T q ) ∪ {v p ,v q } {v p ,v q }
E(T p ⊙ T q )=E(T p ) ⊎ E(T q ) ∪ {(l p ,v p ), (v p ,v q ), (v q ,f q )}
T p ⊙ T q
⊆ /
(T 1 ,c 1 ,k), (T 2 ,c 2 ,k),...(T t ,c t ,k)
T
T = T 1 ⊙ T 2 ⊙ ···⊙ T t .
N ⊙
T N T i
T T(T i )
N T i
T i T i
c T u ∈ T i c(u) =c i (u)
u T i c(u) =c(v) v T i u
(T,k)
∃i, i ∈ [t] T i
T
T i T

T R
T V(R) T(T i ) T(T j )
i ≠ j
V(R) T i T j
T i T j
V(R) T(T i ) i 1 i t R
T i
⊆ /
⊆ /
n O(k 2 )
G =(V, E) k ∈ N c : V → [k] r ∈ V
k
G T k r c
T

T =(V, E) (T, k, c, u)
T T u
O(k 2 ) T u
T T k u
T u
k u
T u O(k 2 ) k
u T 2k
k u
2k u
k u T u
2k k
2k T u
2k+2k·k+2k = O(k 2 ) n u T
n
O(k 2 )

(T, k, c) T {v 1 ,v 2 ,...,v l }
T 1 T l T 1 ,T 2 ,...,T l T
v 1 i l v v i T i
T ′ c ′ T
T k + 1 v (T ′ ,k+ 1, c ′ ,v)
T ′
(T, k, c) (T ′ ,k+ 1, c ′ ,v)
⊆ /
(T 1 ,v 1 ,c 1 ,k), (T 2 ,v 2 ,c 2 ,k),...(T r ,v r ,c r ,k) T i =
(V i ,E i ) T =(V, E)
V = {u} ∪ {u 1 ,u 2 ,...,u r }∪ i∈[r] V i
(u, u i ) ∈ E i ∈ [r] (u i ,v i ) ∈ E i ∈ [r]
E i ⊂ E i ∈ [t]
c : V → [k + 2] v ∈
V i c(v) =c i (v) c(u i )=k + 1 u i c(u) =k + 2
(T,k + 2, c, v) T i
v i k + 2

v v i (v i ,u i ) (u i ,u)
T T ′ u
T ′ T i T ′
T i T j T ′ u i u j
T ′
G =(V, E) c : V → [k]
U ⊆ V |U| = s = O(1)
k
G T k U ⊆ V(T) c
T
⊆ /

(T, v, c, k) T =(V, E)
s ′ = s − 1 T ′ =(V ′ ,E ′ ) T ′ =(V ′ ,E ′ )
V ′ = V ∪ {u 1 ,u 2 ,...,u ′ s} E ′ = E ∪ {(v, u 1 ),...,(v, u ′ s)}
c ′ : V ′ → [k + s ′ ]
c ′ (v) =c(v) ∀v ∈ V
c ′ (u j )=k + j ∀j ∈ [s ′ ]
(T ′ ,c ′ , {v, u 1 ,...,u ′ s},k+ s ′ )
(G, k, c)
C(i) C i
C(i) ={v ∈ G | c(v) =i}. C(i) i
(H, k)
S S ⊆ V (u, v) u, v ∈ S u ≠
v

(S, T) S, T ⊆ V S ∩ T = ∅
(u, v) u ∈ S v ∈ T
(S, T) S, T ⊆ V S ∩ T = ∅ |S| = |T|
S T S = {s 1 ,s 2 ,...,s r }
T = {t 1 ,t 2 ,...,t r } (s i ,t i ) i 1 i r
u S u S ⊆ V (u, v)
v ∈ S
H G H
G H
H = G
C(i) i ∈ [k]
C(i) (k + 1) v i (1),v i (2),...,v i (k + 1)
H(i) (C(i),H(i))
(i, j) i ≠ j (H(i),H(j))
H i H j
j th H i v i (j)
(v i (p),v j (q)) p ≠ q
D(i, j) D = ∪ P D(i, j) P = ( )
[k]
2
D
u ∈ D(i, j) u C(l) l l ≠ i l ≠
j
H
u w
u ∈ C(i) w ∈ C(i) u w
C(i)

u ∈ C(i) w ∈ C(j) j ≠ i
u ∈ H(i) w ∈ H(i)
x ∈ C(i) (u, x) (w, x)
u ∈ H(i) v ∈ H(j) i ≠ j u = v i (p) w = v j (q) p = q
u v
u w
u ∈ H(i) w ∈ C(i) (u, w)
u ∈ H(i) w ∈ C(j) j ≠ i u = v i (p) (u, v j (p))

C H D
C G D
D
H D
D D
D
(v j (p),w)
u ∈ D w ∈ D u w D
u ∈ D(i, j) w ∈ C(l) l ≠ i l ≠ j
u w u C(l)
u ∈ D(i, j) w ∈ C(i) w ∈ C(j) u
H(i) H(j) w
u ∈ D w ∈ H(j) j u w
w w D
D
H
G H
H(i) i ∈ [k]
H(i) C(i)
C(i) D
k G

C(i) S
k
|S ∩ C(i)| 1.
|S| k |S ∩ C(i)| = 1 S
S |S ∩
C(i)| 1
|S ∩ C(i)| = 0
u/∈ C(i) H(i)
u ∈ H(i) H(i) u
u ∈ D u H(i)
H(i)
u ∈ H(j) j ≠ i u H(i)
H(i)
H(i)
C(i)
k H(i) S
k

C(i) S
k
|S ∩ C(i)| 1.
|S| k |S ∩ C(i)| = 1 S
S |S ∩
C(i)| 1
|S ∩ C(i)| = 0
u/∈ C(i) H(i)
u ∈ H(i) H(i) u
u ∈ D u H(i)
H(i)
u ∈ H(j) j ≠ i u H(i)
H(i)
H(i)
C(i)
k H(i) S
k

F
F
F
F
p p
F
♦ F
♦ F = {O 1 ,...,O p }
♦ F = {K 5 ,K 3,3 }
F F

⋆ F O(log 3/2 OPT)
F
⋆ t
K 1,t F
F
⋆ F θ c
c F
⋆ F
F
F
F
⋆
F 2 O(k log k) n O(1)
(2 kO(1) n O(1) )
P P
P
G
H O( √ |V(G)|)
F F
O( √ n)

F
♦ F
F
♦ F F
n ( 1 2 −ε) ε >0
F
F
F
F
F = { θ c }
F
θ c
θ c
k
k θ c
k
k
k
♦ H M(H)
♦ H M(H)

n
⋆
O(k 2 ) n
O(k 2 )
⋆
⊆ /

n
⋆
O(k 2 ) n
O(k 2 )
⋆
⊆ /

n
⋆
O(k 2 ) n
O(k 2 )
⋆
⊆ /

+
2
+
+

+
2
+
+

k O(n 2 )
+
2

W[1]
H
k

+
+

+
+
+
H

O( √ |V||E|)

O( √ |V||E|)

O( √ |V||E|)