MATH10200705001 Neeldhara Misra - Homi Bhabha National ...

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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
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KERNELS FOR THE F-DELETION PROBLEM
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DECLARATION I, hereby declare that
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I will be forever grateful to all t
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parameter tractable for every finit
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d
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F P {F}
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F F
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k
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c c = 5 c c = 5
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F i C(F i ) Q Q
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F i C(F i ) Q Q
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φ |x| k |x ′ | |x| L
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H H
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Deletion problem, Deletion we are a
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G M S G S
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+ +
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G H F H J k
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V(G)
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T i a, b, c V(T 1 )
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G χ(G) (G) +1 χ(G) G
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∨ ∧ ¬ ⇔ ⇒ ∀
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φ k φ 3 n
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C R N[C] ∩ R = ∅ E ′
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{u, v} N[u] =N[v]
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q q
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v ∈ A C(v) v v C(v)
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G A q H q = 1 ⎧
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G M ′ G M ′ k χ(G)
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⎫⎪ ⎬ ⎪ ⎭ N(v)
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Q S ⊆ A, N(T) ⊆ S T ⊆ B
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K 4 W n n
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⊲ (G, k)
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(G, k) u 1 ,u 2 ,...,u r
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→ G x x x k
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H r H
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C H v u
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S v Z v u v G
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v T W k X ⊆ W
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O(k 3 ) O(k 3
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|V(H)| H G \ H H
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t ⊕
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Π G t S t S
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f G T = {g | g : 2 X → {0, 1,
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Π n 1 G n V(G n
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t G S ⊆ V(G) (G ′ ,S
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X ′ b b b cc H
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⎧ ⎪⎨ V i V j V 1 V 2 ··
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v i y x ⎨ ⎩
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T \ S ′ |X(T i ′)| s T i v
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G \ X ∂(X) ={x, y, z} x z X
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C i A v v
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θ c θ c
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Y u ∈ X u ∈ Y u y
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F 20 2l5 (l×l) S
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F P K i,j
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F P ♦ H h (t × t)
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F P t : {b, x, y} {b, x, y}
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F P (G, k) (G) d S F
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F P (G, k) α (G) α
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F P G ∗ 0 H
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F P ⊲ µ(G)
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F P t r s t
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F P d ′ l > (k + d)d
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F P β
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F P k j (1)+k j (2)+...+ k j (2
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F P (k √ log(k + d)) · (lo
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F P
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{F}
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{F} ⎫ S } F |S| = (k) X (X
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{F} F
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{F} G k k G
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Θ C O(k 2−ε ) coNP ⊆
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Θ C • t s t H t H s
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Θ C
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Θ C (G, k) Θ c (G, k)
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Θ C v H v θ c
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Θ C v T ⊆ D c c = 5 S
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Θ C G R \{v} G \{v}
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Θ C (G, k) X := (G, k) G, k
Page 161 and 162: Θ C O(k 2 log 3/2 k)
Page 163 and 164: F G \ S S G \ S G[S] F
Page 165 and 166: F {a, b, c, d,
Page 167 and 168: F s p s
Page 169 and 170: F e h H c[e] C
Page 171 and 172: F F t G 1 G 2 G 1
Page 173 and 174: F G 1 ⊕G 3 \Z F
Page 175 and 176: F Q ∩ X i ⊆ ∂(X i ) Q
Page 177 and 178: F O(n + m) =O(n 2 )
Page 179 and 180: M θC ♦ H = Mθ 2
Page 181 and 182: M θC u ∈ B i v ∈ B j y
Page 183 and 184: M θC v 1 v 2 ··· v s v s+1 ··
Page 185 and 186: M θC G A G B X
Page 187 and 188: M θC 2c 2
Page 189 and 190: M θC S(G, µ(G)
Page 191 and 192: M θC v S i ∩ S j = {v}, ∀
Page 193 and 194: M θC u i x i u i
Page 195 and 196: M θC
Page 197 and 198: D(x) ∈ Q ′ i ∈ [t] :x i
Page 199 and 200: P Q P Q P ppt Q
Page 201 and 202: k k k
Page 203 and 204: + k n O(k 3 )
Page 205 and 206: c = 1 θ c k G
Page 207 and 208: θ c c = 5 i v j v
Page 209 and 210: w L k (w) ≠ l w 2 M
Page 211: w = 1122312 ← y 2 1 1 2 2 3
Page 215 and 216: ⊆ /
Page 217 and 218: ⊆ /
Page 219 and 220: R ⊆ F ⋃ S∈R S = U
Page 221 and 222: F i r r u i T
Page 223 and 224: k k (T, k, c)
Page 225 and 226: (T, k, c) (Q, c q ,k)
Page 227 and 228: R R T(i)
Page 229 and 230: T p T q l p ∈ V(T p )
Page 231 and 232: T =(V, E) (T, k, c, u)
Page 233 and 234: v v i (v i ,u i ) (u i ,u
Page 235 and 236: (S, T) S, T ⊆ V S ∩ T = ∅
Page 237 and 238: C H D C G D D H
Page 239 and 240: C(i) S k |S ∩ C(i)|
Page 241 and 242: ⋆ F O(log 3/2 OPT) F
Page 243 and 244: n
Page 245 and 246: n
Page 247 and 248: +
Page 249 and 250: W[1]
Page 251 and 252: + + +
Page 253 and 254: O( √ |V||E|)
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MATH10200705001 Neeldhara Misra - Homi Bhabha National ...

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