MATH10200705001 Neeldhara Misra - Homi Bhabha National ...

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{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
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 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 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 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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MATH10200705001 Neeldhara Misra - Homi Bhabha National ...

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