MATH10200705001 Neeldhara Misra - Homi Bhabha National ...

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(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
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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
Page 123 and 124: F P ⊲ µ(G)
Page 125 and 126: F P t r s t
Page 127 and 128: F P d ′ l > (k + d)d
Page 129 and 130: F P β
Page 131 and 132: F P k j (1)+k j (2)+...+ k j (2
Page 133 and 134: F P (k √ log(k + d)) · (lo
Page 135 and 136: F P
Page 137 and 138: {F}
Page 139 and 140: {F} ⎫ S } F |S| = (k) X (X
Page 141 and 142: {F} F
Page 143 and 144: {F} G k k G
Page 145 and 146: Θ C O(k 2−ε ) coNP ⊆
Page 147 and 148: Θ C • t s t H t H s
Page 149 and 150: Θ C
Page 151 and 152: Θ C (G, k) Θ c (G, k)
Page 153 and 154: Θ C v H v θ c
Page 155 and 156: Θ C v T ⊆ D c c = 5 S
Page 157 and 158: Θ C G R \{v} G \{v}
Page 159 and 160: 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: F G 1 ⊕G 3 \Z F
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 and 212: w = 1122312 ← y 2 1 1 2 2 3
Page 213 and 214: C i (x p ,y p ) C j x
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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
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(T, k, c) (Q, c q ,k)
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R R T(i)
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T p T q l p ∈ V(T p )
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T =(V, E) (T, k, c, u)
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v v i (v i ,u i ) (u i ,u
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(S, T) S, T ⊆ V S ∩ T = ∅
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C H D C G D D H
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C(i) S k |S ∩ C(i)|
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⋆ F O(log 3/2 OPT) F
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n
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n
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+
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W[1]
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+ + +
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O( √ |V||E|)
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MATH10200705001 Neeldhara Misra - Homi Bhabha National ...

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