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P s rt r r strts t r t srs r

  1. ❙tr✉❝t✉r❡ ❙❡t ❝♦✈❡r ✐s ❛ ✈❡r② ❣❡♥❡r❛❧ ♣r♦❜❧❡♠✱ ✇❡ ❝❛♥ ♦❢t❡♥ r❡str✐❝t ♦✉rs❡❧✈❡s t♦ ❛ ♠♦r❡ ❝♦♥str❛✐♥t str✉❝t✉r❡✳ ●r❛♣❤ str✉❝t✉r❡ ●❡♦♠❡tr✐❝ str✉❝t✉r❡ • ❤②♣❡r❡❞❣❡s ❛r❡ ❡❞❣❡s ♦❢ t❤❡ ❣r❛♣❤ ✭❡❞❣❡ ❝♦✈❡r✮ • ❤②♣❡r❡❞❣❡s ❛r❡ ❝❧♦s❡❞ ♥❡✐❣❤❜♦r❤♦♦❞s • ❤②♣❡r❡❞❣❡s ❛r❡ s❡ts ♦❢ ♣♦✐♥ts t❤❛t ♦❢ t❤❡ ❣r❛♣❤ ✭❞♦♠✐♥❛t✐♦♥✮ ❝❛♥ ❜❡ ❝♦✈❡r❡❞ ❜② t❤❡ s❛♠❡ ❞✐s❦ ✺✴✹✾

  2. ❙tr♦♥❣ ❣❡♦❞❡t✐❝ ♥✉♠❜❡r ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ♣♦✐♥ts ✉s✐♥❣ ❞✐s❦s ▼② P❤❉ ❙❡t ❝♦✈❡r ❙tr✉❝t✉r❡✿ ❱❛r✐❛t✐♦♥✿ ●r❛♣❤s • • Pr♦♣❛❣❛t✐♦♥ ●❡♦♠❡tr✐❝ • • ●❛♠❡ • ■❞❡♥t✐✜❝❛t✐♦♥ ❙tr♦♥❣ ❣❡♦❞❡t✐❝ ♥✉♠❜❡r ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ♣♦✐♥ts ✉s✐♥❣ ❞✐s❦s ✻✴✹✾

  3. ❙tr♦♥❣ ❣❡♦❞❡t✐❝ ♥✉♠❜❡r ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ♣♦✐♥ts ✉s✐♥❣ ❞✐s❦s ▼② P❤❉ ❙❡t ❝♦✈❡r ❙tr✉❝t✉r❡✿ ❱❛r✐❛t✐♦♥✿ ●r❛♣❤s • • Pr♦♣❛❣❛t✐♦♥ ●❡♦♠❡tr✐❝ • • ●❛♠❡ • ■❞❡♥t✐✜❝❛t✐♦♥ ❙tr♦♥❣ ❣❡♦❞❡t✐❝ ♥✉♠❜❡r ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ♣♦✐♥ts ✉s✐♥❣ ❞✐s❦s ✻✴✹✾

  4. ▼❛❦❡r✲❇r❡❛❦❡r ❉♦♠✐♥❛t✐♦♥ ●❛♠❡ ✼✴✹✾

  5. ❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s ✭❖r❡✱ ✶✾✻✶✮ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ❜❡ ❛ s✉❜s❡t ♦❢ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S ✳ ✽✴✹✾

  6. ❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s ✭❖r❡✱ ✶✾✻✶✮ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ❜❡ ❛ s✉❜s❡t ♦❢ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S ✳ ✽✴✹✾

  7. ❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s ✭❖r❡✱ ✶✾✻✶✮ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ❜❡ ❛ s✉❜s❡t ♦❢ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S ✳ ✽✴✹✾

  8. ❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s ✭❖r❡✱ ✶✾✻✶✮ ▲❡t G = ( V , E ) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ❜❡ ❛ s✉❜s❡t ♦❢ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S ✳ ✽✴✹✾

  9. ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r ❆❧t❡r♥❛t❡❧② s❡❧❡❝t ❛ ✈❡rt❡① ♦❢ t❤❡ ❣r❛♣❤ t❤❛t ❞♦♠✐♥❛t❡s ❛t ❧❡❛st ♦♥❡ ♥❡✇ ✈❡rt❡① ❉♦♠✐♥❛t♦r ✇❛♥ts t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡t t♦ ❜❡ s♠❛❧❧ ❙t❛❧❧❡r ✇❛♥ts ✐t t♦ ❜❡ ❧❛r❣❡ ❉❡t❡r♠✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♠♦✈❡s ✐♥ ❛♥ ♦♣t✐♠❛❧ ❣❛♠❡ ♦❢ t❤❡ ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s ✲❝♦♠♣❧❡t❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✻✮✳ ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡s ●❛♠❡ ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ❉✐s❥♦✐♥t ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ✭❆❧♦♥ ❡t ❛❧✳✱ ✷✵✵✷✮ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ✭❇✉❥tás ❡t ❚✉③❛✱ ✷✵✶✹✮ ✾✴✹✾

  10. ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r ❆❧t❡r♥❛t❡❧② s❡❧❡❝t ❛ ✈❡rt❡① ♦❢ t❤❡ ❣r❛♣❤ t❤❛t ❞♦♠✐♥❛t❡s ❛t ❧❡❛st ♦♥❡ ♥❡✇ ✈❡rt❡① ❉♦♠✐♥❛t♦r ✇❛♥ts t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡t t♦ ❜❡ s♠❛❧❧ ❙t❛❧❧❡r ✇❛♥ts ✐t t♦ ❜❡ ❧❛r❣❡ ❉❡t❡r♠✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♠♦✈❡s ✐♥ ❛♥ ♦♣t✐♠❛❧ ❣❛♠❡ ♦❢ t❤❡ ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s ✲❝♦♠♣❧❡t❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✻✮✳ ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡s ●❛♠❡ ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ❉✐s❥♦✐♥t ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ✭❆❧♦♥ ❡t ❛❧✳✱ ✷✵✵✷✮ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ✭❇✉❥tás ❡t ❚✉③❛✱ ✷✵✶✹✮ ✾✴✹✾

  11. ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ❉❡t❡r♠✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♠♦✈❡s ✐♥ ❛♥ ♦♣t✐♠❛❧ ❣❛♠❡ ♦❢ t❤❡ ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s ✲❝♦♠♣❧❡t❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✻✮✳ ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡s ●❛♠❡ ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ❉✐s❥♦✐♥t ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ✭❆❧♦♥ ❡t ❛❧✳✱ ✷✵✵✷✮ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ✭❇✉❥tás ❡t ❚✉③❛✱ ✷✵✶✹✮ • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❆❧t❡r♥❛t❡❧② s❡❧❡❝t ❛ ✈❡rt❡① ♦❢ t❤❡ ❣r❛♣❤ t❤❛t ❞♦♠✐♥❛t❡s ❛t ❧❡❛st ♦♥❡ ♥❡✇ ✈❡rt❡① • ❉♦♠✐♥❛t♦r ✇❛♥ts t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡t t♦ ❜❡ s♠❛❧❧ • ❙t❛❧❧❡r ✇❛♥ts ✐t t♦ ❜❡ ❧❛r❣❡ ✾✴✹✾

  12. ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ❉❡t❡r♠✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♠♦✈❡s ✐♥ ❛♥ ♦♣t✐♠❛❧ ❣❛♠❡ ♦❢ t❤❡ ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s ✲❝♦♠♣❧❡t❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✻✮✳ ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡s ●❛♠❡ ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ❉✐s❥♦✐♥t ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ✭❆❧♦♥ ❡t ❛❧✳✱ ✷✵✵✷✮ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ✭❇✉❥tás ❡t ❚✉③❛✱ ✷✵✶✹✮ • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❆❧t❡r♥❛t❡❧② s❡❧❡❝t ❛ ✈❡rt❡① ♦❢ t❤❡ ❣r❛♣❤ t❤❛t ❞♦♠✐♥❛t❡s ❛t ❧❡❛st ♦♥❡ ♥❡✇ ✈❡rt❡① • ❉♦♠✐♥❛t♦r ✇❛♥ts t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡t t♦ ❜❡ s♠❛❧❧ • ❙t❛❧❧❡r ✇❛♥ts ✐t t♦ ❜❡ ❧❛r❣❡ D ✾✴✹✾

  13. ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ❉❡t❡r♠✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♠♦✈❡s ✐♥ ❛♥ ♦♣t✐♠❛❧ ❣❛♠❡ ♦❢ t❤❡ ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s ✲❝♦♠♣❧❡t❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✻✮✳ ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡s ●❛♠❡ ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ❉✐s❥♦✐♥t ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ✭❆❧♦♥ ❡t ❛❧✳✱ ✷✵✵✷✮ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ✭❇✉❥tás ❡t ❚✉③❛✱ ✷✵✶✹✮ • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❆❧t❡r♥❛t❡❧② s❡❧❡❝t ❛ ✈❡rt❡① ♦❢ t❤❡ ❣r❛♣❤ t❤❛t ❞♦♠✐♥❛t❡s ❛t ❧❡❛st ♦♥❡ ♥❡✇ ✈❡rt❡① • ❉♦♠✐♥❛t♦r ✇❛♥ts t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡t t♦ ❜❡ s♠❛❧❧ • ❙t❛❧❧❡r ✇❛♥ts ✐t t♦ ❜❡ ❧❛r❣❡ D S ✾✴✹✾

  14. ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ❉❡t❡r♠✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♠♦✈❡s ✐♥ ❛♥ ♦♣t✐♠❛❧ ❣❛♠❡ ♦❢ t❤❡ ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s ✲❝♦♠♣❧❡t❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✻✮✳ ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡s ●❛♠❡ ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ❉✐s❥♦✐♥t ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ✭❆❧♦♥ ❡t ❛❧✳✱ ✷✵✵✷✮ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ✭❇✉❥tás ❡t ❚✉③❛✱ ✷✵✶✹✮ • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❆❧t❡r♥❛t❡❧② s❡❧❡❝t ❛ ✈❡rt❡① ♦❢ t❤❡ ❣r❛♣❤ t❤❛t ❞♦♠✐♥❛t❡s ❛t ❧❡❛st ♦♥❡ ♥❡✇ ✈❡rt❡① • ❉♦♠✐♥❛t♦r ✇❛♥ts t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡t t♦ ❜❡ s♠❛❧❧ • ❙t❛❧❧❡r ✇❛♥ts ✐t t♦ ❜❡ ❧❛r❣❡ D S D ✾✴✹✾

  15. ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡s ●❛♠❡ ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ❉✐s❥♦✐♥t ❞♦♠✐♥❛t✐♦♥ ♥✉♠❜❡r ✭❆❧♦♥ ❡t ❛❧✳✱ ✷✵✵✷✮ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✵✮ ✭❇✉❥tás ❡t ❚✉③❛✱ ✷✵✶✹✮ • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❆❧t❡r♥❛t❡❧② s❡❧❡❝t ❛ ✈❡rt❡① ♦❢ t❤❡ ❣r❛♣❤ t❤❛t ❞♦♠✐♥❛t❡s ❛t ❧❡❛st ♦♥❡ ♥❡✇ ✈❡rt❡① • ❉♦♠✐♥❛t♦r ✇❛♥ts t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡t t♦ ❜❡ s♠❛❧❧ • ❙t❛❧❧❡r ✇❛♥ts ✐t t♦ ❜❡ ❧❛r❣❡ D S D ❉❡t❡r♠✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♠♦✈❡s ✐♥ ❛♥ ♦♣t✐♠❛❧ ❣❛♠❡ ♦❢ t❤❡ ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s PSPACE ✲❝♦♠♣❧❡t❡ ✭❇r❡➨❛r ❡t ❛❧✳✱ ✷✵✶✻✮✳ ✾✴✹✾

  16. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ ✶✵✴✹✾

  17. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ D ✶✵✴✹✾

  18. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ × D S ✶✵✴✹✾

  19. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ × D S D ✶✵✴✹✾

  20. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ × D S D ✶✵✴✹✾

  21. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ × D S D ✶✵✴✹✾

  22. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ ✶✵✴✹✾

  23. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ × S ✶✵✴✹✾

  24. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ D × S ✶✵✴✹✾

  25. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ D × × S S ✶✵✴✹✾

  26. ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉ ❛♥❞ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ • P❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) • ❚✇♦ ♣❧❛②❡rs✿ ❉♦♠✐♥❛t♦r ❛♥❞ ❙t❛❧❧❡r • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ V ✳ • ❉♦♠✐♥❛t♦r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ✈❡rt✐❝❡s t❤❛t ❤❡ s❡❧❡❝t❡❞ ✐♥❞✉❝❡ ❛ ❞♦♠✐♥❛t✐♥❣ s❡t✳ D × × S S ✶✵✴✹✾

  27. ❚❤❡ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❙t❛❧❧❡r st❛rts ❉♦♠✐♥❛t♦r ✇✐♥s ❙t❛❧❧❡r ✇✐♥s ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ✇✐♥s ❙t❛❧❧❡r ✇✐♥s ❚❤❡ ♣r♦❜❧❡♠ ❚❤❡ ❣♦❛❧ ✐s t♦ ❞❡❝✐❞❡ ✇❤✐❝❤ ♣❧❛②❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣②✳ ✶✶✴✹✾

  28. ❚❤❡ ♣r♦❜❧❡♠ ❚❤❡ ❣♦❛❧ ✐s t♦ ❞❡❝✐❞❡ ✇❤✐❝❤ ♣❧❛②❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣②✳ ❚❤❡ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❙t❛❧❧❡r st❛rts ❉♦♠✐♥❛t♦r ✇✐♥s ❙t❛❧❧❡r ✇✐♥s ❉♦♠✐♥❛t♦r st❛rts D N ❉♦♠✐♥❛t♦r ✇✐♥s ❙t❛❧❧❡r ✇✐♥s P S ✶✶✴✹✾

  29. ❚❤❡ ▼❛❦❡r✲❇r❡❛❦❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s ❛ ▼❛❦❡r ❇r❡❛❦❡r ❣❛♠❡✳ ❂ ④t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡ts⑥✱ ❉♦♠✐♥❛t♦r ❂ ▼❛❦❡r✳ ❂ ④t❤❡ ❝❧♦s❡❞ ♥❡✐❣❤❜♦r❤♦♦❞s⑥✱ ❙t❛❧❧❡r ❂ ▼❛❦❡r✳ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s • P❧❛②❡❞ ♦♥ ❛♥ ❤②♣❡r❣r❛♣❤ ( X , F ) ✳ • ❚✇♦ ♣❧❛②❡rs✿ ▼❛❦❡r ❛♥❞ ❇r❡❛❦❡r✳ • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ X ✳ • ▼❛❦❡r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❤❡ s❡❧❡❝t❡❞ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❤②♣❡r❡❞❣❡ A ∈ F ✳ • ❍❡① ◮ ▼❛❦❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✭◆❛s❤✱ ✶✾✺✷✮ ✶✷✴✹✾

  30. ❂ ④t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡ts⑥✱ ❉♦♠✐♥❛t♦r ❂ ▼❛❦❡r✳ ❂ ④t❤❡ ❝❧♦s❡❞ ♥❡✐❣❤❜♦r❤♦♦❞s⑥✱ ❙t❛❧❧❡r ❂ ▼❛❦❡r✳ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s • P❧❛②❡❞ ♦♥ ❛♥ ❤②♣❡r❣r❛♣❤ ( X , F ) ✳ • ❚✇♦ ♣❧❛②❡rs✿ ▼❛❦❡r ❛♥❞ ❇r❡❛❦❡r✳ • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ X ✳ • ▼❛❦❡r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❤❡ s❡❧❡❝t❡❞ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❤②♣❡r❡❞❣❡ A ∈ F ✳ • ❍❡① ◮ ▼❛❦❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✭◆❛s❤✱ ✶✾✺✷✮ ❚❤❡ ▼❛❦❡r✲❇r❡❛❦❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s ❛ ▼❛❦❡r ❇r❡❛❦❡r ❣❛♠❡✳ ✶✷✴✹✾

  31. ❂ ④t❤❡ ❝❧♦s❡❞ ♥❡✐❣❤❜♦r❤♦♦❞s⑥✱ ❙t❛❧❧❡r ❂ ▼❛❦❡r✳ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s • P❧❛②❡❞ ♦♥ ❛♥ ❤②♣❡r❣r❛♣❤ ( X , F ) ✳ • ❚✇♦ ♣❧❛②❡rs✿ ▼❛❦❡r ❛♥❞ ❇r❡❛❦❡r✳ • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ X ✳ • ▼❛❦❡r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❤❡ s❡❧❡❝t❡❞ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❤②♣❡r❡❞❣❡ A ∈ F ✳ • ❍❡① ◮ ▼❛❦❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✭◆❛s❤✱ ✶✾✺✷✮ ❚❤❡ ▼❛❦❡r✲❇r❡❛❦❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s ❛ ▼❛❦❡r ❇r❡❛❦❡r ❣❛♠❡✳ • F ❂ ④t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡ts⑥✱ ❉♦♠✐♥❛t♦r ❂ ▼❛❦❡r✳ ✶✷✴✹✾

  32. ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s • P❧❛②❡❞ ♦♥ ❛♥ ❤②♣❡r❣r❛♣❤ ( X , F ) ✳ • ❚✇♦ ♣❧❛②❡rs✿ ▼❛❦❡r ❛♥❞ ❇r❡❛❦❡r✳ • ❚❤❡② ❛❧t❡r♥❛t❡❧② s❡❧❡❝t ✈❡rt✐❝❡s ♦❢ X ✳ • ▼❛❦❡r ✇✐♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❤❡ s❡❧❡❝t❡❞ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❤②♣❡r❡❞❣❡ A ∈ F ✳ • ❍❡① ◮ ▼❛❦❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✭◆❛s❤✱ ✶✾✺✷✮ ❚❤❡ ▼❛❦❡r✲❇r❡❛❦❡r ❉♦♠✐♥❛t✐♦♥ ❣❛♠❡ ✐s ❛ ▼❛❦❡r ❇r❡❛❦❡r ❣❛♠❡✳ • F ❂ ④t❤❡ ❞♦♠✐♥❛t✐♥❣ s❡ts⑥✱ ❉♦♠✐♥❛t♦r ❂ ▼❛❦❡r✳ • F ❂ ④t❤❡ ❝❧♦s❡❞ ♥❡✐❣❤❜♦r❤♦♦❞s⑥✱ ❙t❛❧❧❡r ❂ ▼❛❦❡r✳ ✶✷✴✹✾

  33. ■t ✐s ♥❡✈❡r ✐♥t❡r❡st✐♥❣ t♦ ♣❛ss ❚❤❡r❡ ✐s ♥♦ ❣❛♠❡ ♦❢ ♦✉t❝♦♠❡ ❚❤❡♦r❡♠ ✭❙❝❤❛❡❢❡r✱ ✶✾✼✽✮ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ▼❛❦❡r✲❇r❡❛❦❡r ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ❚❤❡♦r❡♠ ✭❋♦❧❦❧♦r❡✮ ■❢ ▼❛❦❡r ✇✐♥s t❤❡ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡ ♦♥ ( X , F ) ❛s t❤❡ s❡❝♦♥❞ ♣❧❛②❡r✱ t❤❡♥ ❤❡ ❛❧s♦ ✇✐♥s ❛s ✜rst ♣❧❛②❡r✳ ✶✸✴✹✾

  34. ❚❤❡♦r❡♠ ✭❙❝❤❛❡❢❡r✱ ✶✾✼✽✮ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ▼❛❦❡r✲❇r❡❛❦❡r ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ❚❤❡♦r❡♠ ✭❋♦❧❦❧♦r❡✮ ■❢ ▼❛❦❡r ✇✐♥s t❤❡ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡ ♦♥ ( X , F ) ❛s t❤❡ s❡❝♦♥❞ ♣❧❛②❡r✱ t❤❡♥ ❤❡ ❛❧s♦ ✇✐♥s ❛s ✜rst ♣❧❛②❡r✳ → ■t ✐s ♥❡✈❡r ✐♥t❡r❡st✐♥❣ t♦ ♣❛ss → ❚❤❡r❡ ✐s ♥♦ ❣❛♠❡ ♦❢ ♦✉t❝♦♠❡ P ✶✸✴✹✾

  35. ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ❚❤❡♦r❡♠ ✭❋♦❧❦❧♦r❡✮ ■❢ ▼❛❦❡r ✇✐♥s t❤❡ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡ ♦♥ ( X , F ) ❛s t❤❡ s❡❝♦♥❞ ♣❧❛②❡r✱ t❤❡♥ ❤❡ ❛❧s♦ ✇✐♥s ❛s ✜rst ♣❧❛②❡r✳ → ■t ✐s ♥❡✈❡r ✐♥t❡r❡st✐♥❣ t♦ ♣❛ss → ❚❤❡r❡ ✐s ♥♦ ❣❛♠❡ ♦❢ ♦✉t❝♦♠❡ P ❚❤❡♦r❡♠ ✭❙❝❤❛❡❢❡r✱ ✶✾✼✽✮ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ▼❛❦❡r✲❇r❡❛❦❡r ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ✶✸✴✹✾

  36. ❉♦♠✐♥❛t♦r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ ❉♦♠✐♥❛t♦r st❛rts ❙t❛❧❧❡r st❛rts ✶✹✴✹✾

  37. ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ✶✹✴✹✾

  38. ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ × ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ✶✹✴✹✾

  39. ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ × ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ✶✹✴✹✾

  40. ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ × × ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r N ✶✹✴✹✾

  41. ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r N ✶✹✴✹✾

  42. ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r N ✶✹✴✹✾

  43. ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ × ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r N ✶✹✴✹✾

  44. ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ × ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r N ✶✹✴✹✾

  45. ❙t❛❧❧❡r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ × × ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r N D ✶✹✴✹✾

  46. ❙t❛❧❧❡r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r N D ✶✹✴✹✾

  47. ❙t❛❧❧❡r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r N D ✶✹✴✹✾

  48. ❙t❛❧❧❡r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ × ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r N D ✶✹✴✹✾

  49. ❙t❛❧❧❡r ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ × ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r N D ✶✹✴✹✾

  50. ❙t❛❧❧❡r ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ × × ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r N D ✶✹✴✹✾

  51. ❖✉t❝♦♠❡s ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ❢♦r t❤❡ t❤r❡❡ ♣♦ss✐❜❧❡s ♦✉t❝♦♠❡s✳ ❉♦♠✐♥❛t♦r st❛rts ❉♦♠✐♥❛t♦r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r ❙t❛❧❧❡r st❛rts ❙t❛❧❧❡r ❉♦♠✐♥❛t♦r ❙t❛❧❧❡r N D S ✶✹✴✹✾

  52. ✹ ✷ ✷ ❆ ✇✐♥♥✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ❉♦♠✐♥❛t♦r ▲❡♠♠❛ ■❢ ❛ ❣r❛♣❤ ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❝❧✐q✉❡s ♦❢ s✐③❡ ❛t ❧❡❛st ✷ t❤❡♥ ✐ts ♦✉t❝♦♠❡ ✐s D ✳ ✶✺✴✹✾

  53. ❆ ✇✐♥♥✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ❉♦♠✐♥❛t♦r ▲❡♠♠❛ ■❢ ❛ ❣r❛♣❤ ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❝❧✐q✉❡s ♦❢ s✐③❡ ❛t ❧❡❛st ✷ t❤❡♥ ✐ts ♦✉t❝♦♠❡ ✐s D ✳ K ✹ K ✷ K ✷ ✶✺✴✹✾

  54. ❆ ✇✐♥♥✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ❉♦♠✐♥❛t♦r ▲❡♠♠❛ ■❢ ❛ ❣r❛♣❤ ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❝❧✐q✉❡s ♦❢ s✐③❡ ❛t ❧❡❛st ✷ t❤❡♥ ✐ts ♦✉t❝♦♠❡ ✐s D ✳ K ✹ K ✷ K ✷ × ✶✺✴✹✾

  55. ❆ ✇✐♥♥✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ❉♦♠✐♥❛t♦r ▲❡♠♠❛ ■❢ ❛ ❣r❛♣❤ ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❝❧✐q✉❡s ♦❢ s✐③❡ ❛t ❧❡❛st ✷ t❤❡♥ ✐ts ♦✉t❝♦♠❡ ✐s D ✳ K ✹ K ✷ K ✷ × ✶✺✴✹✾

  56. ❆ ✇✐♥♥✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ❉♦♠✐♥❛t♦r ▲❡♠♠❛ ■❢ ❛ ❣r❛♣❤ ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❝❧✐q✉❡s ♦❢ s✐③❡ ❛t ❧❡❛st ✷ t❤❡♥ ✐ts ♦✉t❝♦♠❡ ✐s D ✳ × K ✹ K ✷ K ✷ × ✶✺✴✹✾

  57. ❆ ✇✐♥♥✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ❉♦♠✐♥❛t♦r ▲❡♠♠❛ ■❢ ❛ ❣r❛♣❤ ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❝❧✐q✉❡s ♦❢ s✐③❡ ❛t ❧❡❛st ✷ t❤❡♥ ✐ts ♦✉t❝♦♠❡ ✐s D ✳ × K ✹ K ✷ K ✷ × ✶✺✴✹✾

  58. ❆ ✇✐♥♥✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ❉♦♠✐♥❛t♦r ▲❡♠♠❛ ■❢ ❛ ❣r❛♣❤ ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❝❧✐q✉❡s ♦❢ s✐③❡ ❛t ❧❡❛st ✷ t❤❡♥ ✐ts ♦✉t❝♦♠❡ ✐s D ✳ × K ✹ K ✷ K ✷ × × ✶✺✴✹✾

  59. ❆ ✇✐♥♥✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ❉♦♠✐♥❛t♦r ▲❡♠♠❛ ■❢ ❛ ❣r❛♣❤ ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❝❧✐q✉❡s ♦❢ s✐③❡ ❛t ❧❡❛st ✷ t❤❡♥ ✐ts ♦✉t❝♦♠❡ ✐s D ✳ × K ✹ K ✷ K ✷ × × ✶✺✴✹✾

  60. ❆ ✇✐♥♥✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ❉♦♠✐♥❛t♦r ▲❡♠♠❛ ■❢ ❛ ❣r❛♣❤ ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❝❧✐q✉❡s ♦❢ s✐③❡ ❛t ❧❡❛st ✷ t❤❡♥ ✐ts ♦✉t❝♦♠❡ ✐s D ✳ × K ✹ K ✷ K ✷ × × ✶✺✴✹✾

  61. P❛✐r✐♥❣ ❞♦♠✐♥❛t✐♥❣ s❡ts ❉❡✜♥✐t✐♦♥ ✭❉✉❝❤ê♥❡✱ ●✳✱ P❛rr❡❛✉✱ ❘❡♥❛✉❧t✱ ✷✵✶✽✰✮ ❆ s❡t ♦❢ ♣❛✐rs ♦❢ ✈❡rt✐❝❡s { ( u ✶ , v ✶ ) , . . . , ( u k , v k ) } ✐s ❛ ♣❛✐r✐♥❣ ❞♦♠✐♥❛t✐♥❣ s❡t ✐❢✿ • ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ❞✐st✐♥❝t✱ • V = � k i = ✶ N [ u i ] ∩ N [ v i ] . G ❤❛s ❛ ♣❛✐r✐♥❣ ❞♦♠✐♥❛t✐♥❣ s❡t = ⇒ G ❤❛s ♦✉t❝♦♠❡ D ✳ ✶✻✴✹✾

  62. P❛✐r✐♥❣ ❞♦♠✐♥❛t✐♥❣ s❡ts ❚❤❡♦r❡♠ ❉❡❝✐❞✐♥❣ ✐❢ ❛ ❣r❛♣❤ ❛❞♠✐ts ❛ ♣❛✐r✐♥❣ ❞♦♠✐♥❛t✐♥❣ s❡t ✐s ❛♥ ◆P✲ ❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ❚❤❡ ♣r♦♦❢ ✉s❡s ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ ❙❆❚✳ c j ✶ c j ✷ c j k · · · e i e i ●❛❞❣❡t ❢♦r ❛ ✈❛r✐❛❜❧❡ ✶✼✴✹✾

  63. P❛✐r✐♥❣ ❞♦♠✐♥❛t✐♥❣ s❡ts ❚❤❡r❡ ❡①✐st ❣r❛♣❤s ♦❢ ♦✉t❝♦♠❡ D t❤❛t ❞♦ ♥♦t ❛❞♠✐t ♣❛✐r✐♥❣ ❞♦♠✐♥❛t✐♥❣ s❡ts✳ ✶✽✴✹✾

  64. ❉♦♠✐♥❛t♦r ❢♦❧❧♦✇s ❇r❡❛❦❡r✬s str❛t❡❣② ❙t❛❧❧❡r ❢♦❧❧♦✇s ▼❛❦❡r✬s str❛t❡❣② ❈♦♠♣❧❡①✐t② ❚❤❡♦r❡♠ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❛ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♣♦✲ s✐t✐♦♥ ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ❚❤✐s r❡s✉❧t ✐s ♣r♦✈❡❞ ❜② r❡❞✉❝t✐♦♥ ❢r♦♠ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ✇❤✐❝❤ ❛r❡ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ✭❙❝❤❛❡✛❡r✱ ✶✾✼✽✮✳ x ✶ x ✷ x ✸ x ✹ X ✷ X ✸ A ✸ A ✶ X ✶ X ✹ A ✷ a ✵ a ✶ a ✵ a ✶ a ✵ a ✶ ✶ ✶ ✷ ✷ ✸ ✸ ✶✾✴✹✾

  65. ❙t❛❧❧❡r ❢♦❧❧♦✇s ▼❛❦❡r✬s str❛t❡❣② ❈♦♠♣❧❡①✐t② ❚❤❡♦r❡♠ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❛ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♣♦✲ s✐t✐♦♥ ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ❚❤✐s r❡s✉❧t ✐s ♣r♦✈❡❞ ❜② r❡❞✉❝t✐♦♥ ❢r♦♠ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ✇❤✐❝❤ ❛r❡ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ✭❙❝❤❛❡✛❡r✱ ✶✾✼✽✮✳ x ✶ x ✷ x ✸ x ✹ X ✷ X ✸ A ✸ A ✶ X ✶ X ✹ A ✷ a ✵ a ✶ a ✵ a ✶ a ✵ a ✶ ✶ ✶ ✷ ✷ ✸ ✸ ❉♦♠✐♥❛t♦r ❢♦❧❧♦✇s ❇r❡❛❦❡r✬s str❛t❡❣② ✶✾✴✹✾

  66. ❙t❛❧❧❡r ❢♦❧❧♦✇s ▼❛❦❡r✬s str❛t❡❣② ❈♦♠♣❧❡①✐t② ❚❤❡♦r❡♠ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❛ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♣♦✲ s✐t✐♦♥ ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ❚❤✐s r❡s✉❧t ✐s ♣r♦✈❡❞ ❜② r❡❞✉❝t✐♦♥ ❢r♦♠ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ✇❤✐❝❤ ❛r❡ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ✭❙❝❤❛❡✛❡r✱ ✶✾✼✽✮✳ x ✶ x ✷ x ✸ x ✹ X ✷ X ✸ A ✸ A ✶ X ✶ X ✹ A ✷ a ✵ a ✶ a ✵ a ✶ a ✵ a ✶ ✶ ✶ ✷ ✷ ✸ ✸ ❉♦♠✐♥❛t♦r ❢♦❧❧♦✇s ❇r❡❛❦❡r✬s str❛t❡❣② ✶✾✴✹✾

  67. ❙t❛❧❧❡r ❢♦❧❧♦✇s ▼❛❦❡r✬s str❛t❡❣② ❈♦♠♣❧❡①✐t② ❚❤❡♦r❡♠ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❛ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♣♦✲ s✐t✐♦♥ ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ❚❤✐s r❡s✉❧t ✐s ♣r♦✈❡❞ ❜② r❡❞✉❝t✐♦♥ ❢r♦♠ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ✇❤✐❝❤ ❛r❡ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ✭❙❝❤❛❡✛❡r✱ ✶✾✼✽✮✳ x ✶ x ✷ x ✸ x ✹ X ✷ X ✸ A ✸ A ✶ X ✶ X ✹ A ✷ a ✵ a ✶ a ✵ a ✶ a ✵ a ✶ ✶ ✶ ✷ ✷ ✸ ✸ ❉♦♠✐♥❛t♦r ❢♦❧❧♦✇s ❇r❡❛❦❡r✬s str❛t❡❣② ✶✾✴✹✾

  68. ❙t❛❧❧❡r ❢♦❧❧♦✇s ▼❛❦❡r✬s str❛t❡❣② ❈♦♠♣❧❡①✐t② ❚❤❡♦r❡♠ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❛ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♣♦✲ s✐t✐♦♥ ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ❚❤✐s r❡s✉❧t ✐s ♣r♦✈❡❞ ❜② r❡❞✉❝t✐♦♥ ❢r♦♠ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ✇❤✐❝❤ ❛r❡ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ✭❙❝❤❛❡✛❡r✱ ✶✾✼✽✮✳ x ✶ x ✷ x ✸ x ✹ × X ✷ X ✸ A ✸ A ✶ X ✶ X ✹ A ✷ a ✵ a ✶ a ✵ a ✶ a ✵ a ✶ ✶ ✶ ✷ ✷ ✸ ✸ ❉♦♠✐♥❛t♦r ❢♦❧❧♦✇s ❇r❡❛❦❡r✬s str❛t❡❣② ✶✾✴✹✾

  69. ❙t❛❧❧❡r ❢♦❧❧♦✇s ▼❛❦❡r✬s str❛t❡❣② ❈♦♠♣❧❡①✐t② ❚❤❡♦r❡♠ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❛ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♣♦✲ s✐t✐♦♥ ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ❚❤✐s r❡s✉❧t ✐s ♣r♦✈❡❞ ❜② r❡❞✉❝t✐♦♥ ❢r♦♠ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ✇❤✐❝❤ ❛r❡ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ✭❙❝❤❛❡✛❡r✱ ✶✾✼✽✮✳ x ✶ x ✷ x ✸ x ✹ × X ✷ X ✸ A ✸ A ✶ X ✶ X ✹ A ✷ a ✵ a ✶ a ✵ a ✶ a ✵ a ✶ ✶ ✶ ✷ ✷ ✸ ✸ ❉♦♠✐♥❛t♦r ❢♦❧❧♦✇s ❇r❡❛❦❡r✬s str❛t❡❣② ✶✾✴✹✾

  70. ❙t❛❧❧❡r ❢♦❧❧♦✇s ▼❛❦❡r✬s str❛t❡❣② ❈♦♠♣❧❡①✐t② ❚❤❡♦r❡♠ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❛ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♣♦✲ s✐t✐♦♥ ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ❚❤✐s r❡s✉❧t ✐s ♣r♦✈❡❞ ❜② r❡❞✉❝t✐♦♥ ❢r♦♠ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ✇❤✐❝❤ ❛r❡ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ✭❙❝❤❛❡✛❡r✱ ✶✾✼✽✮✳ x ✶ x ✷ x ✸ x ✹ × X ✷ X ✸ A ✸ A ✶ X ✶ X ✹ × A ✷ a ✵ a ✶ a ✵ a ✶ a ✵ a ✶ ✶ ✶ ✷ ✷ ✸ ✸ ❉♦♠✐♥❛t♦r ❢♦❧❧♦✇s ❇r❡❛❦❡r✬s str❛t❡❣② ✶✾✴✹✾

  71. ❙t❛❧❧❡r ❢♦❧❧♦✇s ▼❛❦❡r✬s str❛t❡❣② ❈♦♠♣❧❡①✐t② ❚❤❡♦r❡♠ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❛ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♣♦✲ s✐t✐♦♥ ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ❚❤✐s r❡s✉❧t ✐s ♣r♦✈❡❞ ❜② r❡❞✉❝t✐♦♥ ❢r♦♠ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ✇❤✐❝❤ ❛r❡ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ✭❙❝❤❛❡✛❡r✱ ✶✾✼✽✮✳ x ✶ x ✷ x ✸ x ✹ × X ✷ X ✸ A ✸ A ✶ X ✶ X ✹ × A ✷ a ✵ a ✶ a ✵ a ✶ a ✵ a ✶ ✶ ✶ ✷ ✷ ✸ ✸ ❉♦♠✐♥❛t♦r ❢♦❧❧♦✇s ❇r❡❛❦❡r✬s str❛t❡❣② ✶✾✴✹✾

  72. ❉♦♠✐♥❛t♦r ❢♦❧❧♦✇s ❇r❡❛❦❡r✬s str❛t❡❣② ❈♦♠♣❧❡①✐t② ❚❤❡♦r❡♠ ❉❡❝✐❞✐♥❣ t❤❡ ♦✉t❝♦♠❡ ♦❢ ❛ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♣♦✲ s✐t✐♦♥ ✐s ❛ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✳ ❚❤✐s r❡s✉❧t ✐s ♣r♦✈❡❞ ❜② r❡❞✉❝t✐♦♥ ❢r♦♠ ▼❛❦❡r✲❇r❡❛❦❡r ❣❛♠❡s ✇❤✐❝❤ ❛r❡ P❙P❆❈❊✲❝♦♠♣❧❡t❡ ✭❙❝❤❛❡✛❡r✱ ✶✾✼✽✮✳ x ✶ x ✷ x ✸ x ✹ X ✷ X ✸ A ✸ A ✶ X ✶ X ✹ A ✷ a ✵ a ✶ a ✵ a ✶ a ✵ a ✶ ✶ ✶ ✷ ✷ ✸ ✸ ❙t❛❧❧❡r ❢♦❧❧♦✇s ▼❛❦❡r✬s str❛t❡❣② ✶✾✴✹✾

  73. ❚r❡❡s ❄ ❈♦♠♣❧❡①✐t② ●r❛♣❤s P❙P❆❈❊✲❝ ❇✐♣❛rt✐t❡ ❈❤♦r❞❛❧ P❙P❆❈❊✲❝ P❙P❆❈❊✲❝ ❣r❛♣❤s ❣r❛♣❤s ❙♣❧✐t ❣r❛♣❤s P❙P❆❈❊✲❝ P❛t❤s P ✷✵✴✹✾

  74. ❈♦♠♣❧❡①✐t② ●r❛♣❤s P❙P❆❈❊✲❝ ❇✐♣❛rt✐t❡ ❈❤♦r❞❛❧ P❙P❆❈❊✲❝ P❙P❆❈❊✲❝ ❣r❛♣❤s ❣r❛♣❤s ❙♣❧✐t ❣r❛♣❤s ❚r❡❡s ❄ P❙P❆❈❊✲❝ P❛t❤s P ✷✵✴✹✾

  75. ✺ ✸ ■s ✐t st✐❧❧ tr✉❡ ❢♦r ♦t❤❡r ❣r❛♣❤s❄ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♦♥ tr❡❡s ❋♦r ♣❛t❤s✱ r❡♠♦✈✐♥❣ P ✷ ✬s ♣r❡s❡r✈❡s t❤❡ ♦✉t❝♦♠❡✳ P ✻ P ✹ D D ✷✶✴✹✾

  76. ✻ ✹ ■s ✐t st✐❧❧ tr✉❡ ❢♦r ♦t❤❡r ❣r❛♣❤s❄ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♦♥ tr❡❡s ❋♦r ♣❛t❤s✱ r❡♠♦✈✐♥❣ P ✷ ✬s ♣r❡s❡r✈❡s t❤❡ ♦✉t❝♦♠❡✳ P ✺ P ✸ N N ✷✶✴✹✾

  77. ✻ ✹ ▼❛❦❡r✲❇r❡❛❦❡r ❞♦♠✐♥❛t✐♦♥ ❣❛♠❡ ♦♥ tr❡❡s ❋♦r ♣❛t❤s✱ r❡♠♦✈✐♥❣ P ✷ ✬s ♣r❡s❡r✈❡s t❤❡ ♦✉t❝♦♠❡✳ P ✺ P ✸ N N ■s ✐t st✐❧❧ tr✉❡ ❢♦r ♦t❤❡r ❣r❛♣❤s❄ ✷✶✴✹✾

  78. ●❧✉❡ ♦♣❡r❛t♦r ❲❡ ✧❣❧✉❡✧ t✇♦ ❣r❛♣❤s ♦♥ ❛ ✈❡rt❡①✳ u v G H ✷✷✴✹✾

  79. ●❧✉❡ ♦♣❡r❛t♦r ❲❡ ✧❣❧✉❡✧ t✇♦ ❣r❛♣❤s ♦♥ ❛ ✈❡rt❡①✳ G H G u v H ✷✷✴✹✾

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