MATH10200705001 Neeldhara Misra - Homi Bhabha National ...

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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
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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)
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: θ c θ c
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
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Θ C G R \{v} G \{v}
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Θ C (G, k) X := (G, k) G, k
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Θ C O(k 2 log 3/2 k)
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F G \ S S G \ S G[S] F
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F {a, b, c, d,
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F s p s
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F e h H c[e] C
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F F t G 1 G 2 G 1
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F G 1 ⊕G 3 \Z F
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F Q ∩ X i ⊆ ∂(X i ) Q
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F O(n + m) =O(n 2 )
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M θC ♦ H = Mθ 2
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M θC u ∈ B i v ∈ B j y
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M θC v 1 v 2 ··· v s v s+1 ··
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M θC G A G B X
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M θC 2c 2
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M θC S(G, µ(G)
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M θC v S i ∩ S j = {v}, ∀
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M θC u i x i u i
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M θC
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D(x) ∈ Q ′ i ∈ [t] :x i
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P Q P Q P ppt Q
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k k k
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+ k n O(k 3 )
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c = 1 θ c k G
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θ c c = 5 i v j v
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w L k (w) ≠ l w 2 M
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w = 1122312 ← y 2 1 1 2 2 3
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C i (x p ,y p ) C j x
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⊆ /
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⊆ /
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R ⊆ F ⋃ S∈R S = U
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F i r r u i T
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k k (T, k, c)
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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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