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MATH10200705001 Neeldhara Misra - Homi Bhabha National ...
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v<br />
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<br />
<br />
H v<br />
⋆ <br />
⋆ <br />
c c = 5<br />
G H v D D =<br />
{d 1 ,...,d s } d i D i <br />
(v, d i ) w ∈ D i (v, w) ∈ E(G)<br />
<br />
<br />
<br />
c v ch v H v v (c−1)<br />
D i d(v) >ch v + c(c − 1)h v <br />
|D| ch v q q =<br />
c A = H v B = D S ⊆ H v T ⊆ D S |S| c<br />
T N(T) =S <br />
(v, u) u ∈ D i <br />
d i ∈ T c v w w ∈ S <br />
<br />
c v w c <br />
v w c <br />
v <br />
ch v + c(c − 1)h v ,<br />
h v := |H v | H v θ c θ c v<br />
G H v D D = {d 1 ,...,d s }
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
Page 1 and 2: KERNELS FOR THE F-DELETION PROBLEM
Page 3 and 4: DECLARATION I, hereby declare that
Page 5 and 6: I will be forever grateful to all t
Page 7 and 8: parameter tractable for every finit
Page 9 and 10: d
Page 11 and 12: F P {F}
Page 13: F F
Page 16 and 17: k
Page 18 and 19: c c = 5 c c = 5
Page 20 and 21: F i C(F i ) Q Q
Page 22 and 23: F i C(F i ) Q Q
Page 24 and 25: φ |x| k |x ′ | |x| L
Page 26 and 27: H H
Page 28 and 29: Deletion problem, Deletion we are a
Page 30 and 31: G M S G S
Page 32 and 33: + +
Page 34 and 35: G H F H J k
Page 36 and 37: V(G)
Page 38 and 39: T i a, b, c V(T 1 )
Page 40 and 41: G χ(G) (G) +1 χ(G) G
Page 42 and 43: ∨ ∧ ¬ ⇔ ⇒ ∀
Page 44 and 45: φ k φ 3 n
Page 46 and 47: C R N[C] ∩ R = ∅ E ′
Page 48 and 49: {u, v} N[u] =N[v]
Page 50: q q
Page 53 and 54: v ∈ A C(v) v v C(v)
Page 55 and 56: G A q H q = 1 ⎧
Page 57 and 58: G M ′ G M ′ k χ(G)
Page 59 and 60: ⎫⎪ ⎬ ⎪ ⎭ N(v)
Page 61 and 62: Q S ⊆ A, N(T) ⊆ S T ⊆ B
Page 63 and 64: K 4 W n n
Page 65 and 66: ⊲ (G, k)
Page 67 and 68: (G, k) u 1 ,u 2 ,...,u r
Page 69 and 70: → G x x x k
Page 71 and 72: H r H
Page 73 and 74: C H v u
Page 75 and 76: S v Z v u v G
Page 77 and 78: v T W k X ⊆ W
Page 79 and 80: O(k 3 ) O(k 3
Page 81 and 82: |V(H)| H G \ H H
Page 83 and 84: t ⊕
Page 85 and 86: Π G t S t S
Page 87 and 88: f G T = {g | g : 2 X → {0, 1,
Page 89 and 90: Π n 1 G n V(G n
Page 91 and 92: t G S ⊆ V(G) (G ′ ,S
Page 93 and 94: X ′ b b b cc H
Page 95 and 96: ⎧ ⎪⎨ V i V j V 1 V 2 ··
Page 97 and 98: v i y x ⎨ ⎩
Page 99 and 100: T \ S ′ |X(T i ′)| s T i v
Page 101 and 102: G \ X ∂(X) ={x, y, z} x z X
Page 103 and 104: C i A v v Page 105 and 106: θ c θ c Page 107 and 108: Y u ∈ X u ∈ Y u y Page 109 and 110: F 20 2l5 (l×l) S Page 111 and 112: F P K i,j Page 113 and 114: F P ♦ H h (t × t) Page 115 and 116: F P t : {b, x, y} {b, x, y} Page 117 and 118: F P (G, k) (G) d S F Page 119 and 120: F P (G, k) α (G) α Page 121 and 122: 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: Θ C v H v θ c 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 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 and 212: w = 1122312 ← y 2 1 1 2 2 3
Page 213 and 214: C i (x p ,y p ) C j x
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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