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P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ✐♥ tr✐❛♥❣✉❧❛r ❣r✐❞s Pr♦s❡♥❥✐t ❇♦s❡ ✶ ❱❛❧❡♥t✐♥ ●❧❡❞❡❧ ✷ ❈❧❛✐r❡ P❡♥♥❛r✉♥ ✸ ❙❛♥❞❡r ❱❡r❞♦♥s❝❤♦t ✶ ✶ ❙❝❤♦♦❧ ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ❈❛r❧❡t♦♥ ❯♥✐✈❡rs✐t②✱ ❈❛♥❛❞❛ ✷ ▲■❘■❙✱ ❯♥✐✈❡rs✐té ▲②♦♥ ✶✱ ❋r❛♥❝❡ ✸ ▲■❘▼▼✱ ❈◆❘❙ ✫ ❯♥✐✈✳ ▼♦♥t♣❡❧❧✐❡r✱ ❋r❛♥❝❡ ✶✴✷✵

✷ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ ✱ t❤❡ s✐③❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❞♦♠✐♥❛t✐♥❣ s❡t ✐♥ ❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S ✳ ✷✴✷✵

✷ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ ✱ t❤❡ s✐③❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❞♦♠✐♥❛t✐♥❣ s❡t ✐♥ ❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S ✳ ✷✴✷✵

✷ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ ✱ t❤❡ s✐③❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❞♦♠✐♥❛t✐♥❣ s❡t ✐♥ ❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S ✳ ✷✴✷✵

✷ ❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S ✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ γ ( G ) ✱ t❤❡ s✐③❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❞♦♠✐♥❛t✐♥❣ s❡t ✐♥ G ✷✴✷✵

❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S ✳ γ ( G ) = ✷ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ γ ( G ) ✱ t❤❡ s✐③❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❞♦♠✐♥❛t✐♥❣ s❡t ✐♥ G ✷✴✷✵

✶ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N [ S ] ✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ ( uv ) ∈ E ❛♥❞ N [ u ] \ { v } ⊆ M ✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✸✴✷✵

✶ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N [ S ] ✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ ( uv ) ∈ E ❛♥❞ N [ u ] \ { v } ⊆ M ✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✸✴✷✵

✶ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N [ S ] ✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ ( uv ) ∈ E ❛♥❞ N [ u ] \ { v } ⊆ M ✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✸✴✷✵

✶ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N [ S ] ✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ ( uv ) ∈ E ❛♥❞ N [ u ] \ { v } ⊆ M ✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✸✴✷✵

✶ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N [ S ] ✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ ( uv ) ∈ E ❛♥❞ N [ u ] \ { v } ⊆ M ✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✸✴✷✵

✶ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N [ S ] ✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ ( uv ) ∈ E ❛♥❞ N [ u ] \ { v } ⊆ M ✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✸✴✷✵

P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N [ S ] ✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ ( uv ) ∈ E ❛♥❞ N [ u ] \ { v } ⊆ M ✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ γ P ( G ) = ✶ ✸✴✷✵

❘❡❢♦r♠✉❧❛t❡❞ ✐♥ ❣r❛♣❤ t❡r♠s ❛♥❞ ♣r♦✈❡♥ t♦ ❜❡ ✲❝♦♠♣❧❡t❡ ❍❛②♥❡s✱ ❍❡❞❡t♥✐❡♠✐✱ ❍❡❞❡t♥✐❡♠✐ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✷✮ ❙♦❧✈❡❞ ♦♥ sq✉❛r❡ ❣r✐❞s ❛♥❞ ♦t❤❡r ♣r♦❞✉❝ts ♦❢ ♣❛t❤s ❉♦r✐♥❣ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✻✮ ❉♦r❜❡❝✱ ▼♦❧❧❛r❞✱ ❑❧❛✈➸❛r ❛♥❞ ➆♣❛❝❛♣❛♥ ✭✷✵✵✽✮ ❙♦❧✈❡❞ ♦♥ ❤❡①❛❣♦♥❛❧ ❣r✐❞s ❋❡rr❡r♦✱ ❱❛r❣❤❡s❡ ❛♥❞ ❱✐❥❛②❛❦✉♠❛ ✭✷✵✶✶✮ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ • ■♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♠♦♥✐t♦r✐♥❣ ♣♦✇❡r ❣r✐❞s ◮ ▼✐❧✐✱ ❇❛❧❞✇✐♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✵✮ ◮ ❇❛❧❞✇✐♥✱ ▼✐❧✐✱ ❇♦✐s❡♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✸✮ ✹✴✷✵

❙♦❧✈❡❞ ♦♥ sq✉❛r❡ ❣r✐❞s ❛♥❞ ♦t❤❡r ♣r♦❞✉❝ts ♦❢ ♣❛t❤s ❉♦r✐♥❣ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✻✮ ❉♦r❜❡❝✱ ▼♦❧❧❛r❞✱ ❑❧❛✈➸❛r ❛♥❞ ➆♣❛❝❛♣❛♥ ✭✷✵✵✽✮ ❙♦❧✈❡❞ ♦♥ ❤❡①❛❣♦♥❛❧ ❣r✐❞s ❋❡rr❡r♦✱ ❱❛r❣❤❡s❡ ❛♥❞ ❱✐❥❛②❛❦✉♠❛ ✭✷✵✶✶✮ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ • ■♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♠♦♥✐t♦r✐♥❣ ♣♦✇❡r ❣r✐❞s ◮ ▼✐❧✐✱ ❇❛❧❞✇✐♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✵✮ ◮ ❇❛❧❞✇✐♥✱ ▼✐❧✐✱ ❇♦✐s❡♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✸✮ • ❘❡❢♦r♠✉❧❛t❡❞ ✐♥ ❣r❛♣❤ t❡r♠s ❛♥❞ ♣r♦✈❡♥ t♦ ❜❡ NP ✲❝♦♠♣❧❡t❡ ◮ ❍❛②♥❡s✱ ❍❡❞❡t♥✐❡♠✐✱ ❍❡❞❡t♥✐❡♠✐ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✷✮ ✹✴✷✵

P♦✇❡r ❉♦♠✐♥❛t✐♦♥ • ■♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♠♦♥✐t♦r✐♥❣ ♣♦✇❡r ❣r✐❞s ◮ ▼✐❧✐✱ ❇❛❧❞✇✐♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✵✮ ◮ ❇❛❧❞✇✐♥✱ ▼✐❧✐✱ ❇♦✐s❡♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✸✮ • ❘❡❢♦r♠✉❧❛t❡❞ ✐♥ ❣r❛♣❤ t❡r♠s ❛♥❞ ♣r♦✈❡♥ t♦ ❜❡ NP ✲❝♦♠♣❧❡t❡ ◮ ❍❛②♥❡s✱ ❍❡❞❡t♥✐❡♠✐✱ ❍❡❞❡t♥✐❡♠✐ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✷✮ • ❙♦❧✈❡❞ ♦♥ sq✉❛r❡ ❣r✐❞s ❛♥❞ ♦t❤❡r ♣r♦❞✉❝ts ♦❢ ♣❛t❤s ◮ ❉♦r✐♥❣ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✻✮ ◮ ❉♦r❜❡❝✱ ▼♦❧❧❛r❞✱ ❑❧❛✈➸❛r ❛♥❞ ➆♣❛❝❛♣❛♥ ✭✷✵✵✽✮ • ❙♦❧✈❡❞ ♦♥ ❤❡①❛❣♦♥❛❧ ❣r✐❞s ◮ ❋❡rr❡r♦✱ ❱❛r❣❤❡s❡ ❛♥❞ ❱✐❥❛②❛❦✉♠❛ ✭✷✵✶✶✮ ✹✴✷✵

❘❡s✉❧t ♦♥ ❤❡①❛❣♦♥❛❧ s❤❛♣❡❞ ❣r✐❞ H k ❚❤❡♦r❡♠ ▲❡t H k ❜❡ ❛ tr✐❛♥❣✉❧❛r ❣r✐❞ ✇✐t❤ ❛ r❡❣✉❧❛r ❤❡①❛❣♦♥❛❧✲s❤❛♣❡❞ ❜♦r✲ � k � ❞❡r ♦❢ ❧❡♥❣t❤ k − ✶✳ ❚❤❡♥✱ γ P ( H k ) = ✳ ✸ k − ✶ ✺✴✷✵