O problema do subgrafo biconexo mínimo generalizado: algoritmos e formulações

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❖ P❘❖❇▲❊▼❆ ❉❖ ❙❯❇●❘❆❋❖ ❇■❈❖◆❊❳❖

▼❮◆■▼❖ ●❊◆❊❘❆▲■❩❆❉❖✿

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❘❆❋❆❊▲ ❙❆◆❚❖❙ ❈❖❊▲❍❖

❖ P❘❖❇▲❊▼❆ ❉❖ ❙❯❇●❘❆❋❖ ❇■❈❖◆❊❳❖

▼❮◆■▼❖ ●❊◆❊❘❆▲■❩❆❉❖✿

❆▲●❖❘■❚▼❖❙ ❊ ❋❖❘▼❯▲❆➬Õ❊❙

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦ ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡r✲ s✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ❝♦♠♦ r❡q✲ ✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦✳

❖r✐❡♥t❛❞♦r✿ ❆❧❡①❛♥❞r❡ ❙❛❧❧❡s ❞❛ ❈✉♥❤❛

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❝

✷✵✶✷✱ ❘❛❢❛❡❧ ❙❛♥t♦s ❈♦❡❧❤♦✳ ❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳

❈♦❡❧❤♦✱ ❘❛❢❛❡❧ ❙❛♥t♦s

❈✻✼✷♣ ❖ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦ ▼í♥✐♠♦ ●❡♥❡r❛❧✐③❛❞♦✿ ❆❧❣♦r✐t♠♦s ❡ ❋♦r♠✉❧❛çõ❡s ✴ ❘❛❢❛❡❧ ❙❛♥t♦s ❈♦❡❧❤♦✳ ✖ ❇❡❧♦ ❍♦r✐③♦♥t❡✱ ✷✵✶✷

①①✐✐✱ ✽✻ ❢✳ ✿ ✐❧✳ ❀ ✷✾❝♠

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✖ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s

❖r✐❡♥t❛❞♦r✿ ❆❧❡①❛♥❞r❡ ❙❛❧❧❡s ❞❛ ❈✉♥❤❛

✶✳ ❈♦♠♣✉t❛çã♦ ✕ ❚❡s❡s✳ ✷✳ ❚❡♦r✐❛ ❞♦s ❣r❛❢♦s ✕ ❚❡s❡s✳ ✸✳ ❘❡❞❡s ❞❡ ❝♦♠♣✉t❛❞♦r❡s ✕ ❚❡s❡s✳ ✹✳ ❆❧❣♦r✐t♠♦s ❞❡ ❝♦♠♣✉t❛❞♦r ✕ ❚❡s❡s✳ ■✳ ❖r✐❡♥t❛❞♦r✳ ■■✳ ❚ít✉❧♦

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❉❡❞✐❝♦ ❡st❛ ❞✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦ ❛ ♠❡✉s ♣❛✐s✱ ▲ú❝✐❛ ❡ ❆❧❜❡rt♦✱ ❛ ♠✐♥❤❛ ✐r♠ã✱ ❆♥❛✱ ❡ ❛♦s ♠❡✉s ❛♠✐❣♦s ❡ ❛♠✐❣❛s✳

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ✐♠❡♥s❛♠❡♥t❡ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♠❡✉s ♣❛✐s ▲ú❝✐❛ ❡ ❆❧❜❡rt♦✱ ❡ ♠✐♥❤❛ ✐r♠ã✱ ❆♥❛✳ ❙❡♠ ♦ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❞❡❧❡s✱ ❡✉ ♥ã♦ t❡r✐❛ ❝❤❡❣❛❞♦ ❛♦♥❞❡ ❝❤❡❣✉❡✐ ❤♦❥❡✳ ❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♠❡✉s ❛♠✐❣♦s ❡ ❛♠✐❣❛s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❙té❧✐♦ ❍❡♥r✐q✉❡s✱ ➶♥❞❡rs♦♥ ❇✐s♣♦✱ ❆♥❞❡rs♦♥ ●♦♥③❛❣❛✱ ❆❧✐♥❡ ❇❡ss❛✱ ▼❛r✐❛♥❛ P❛♥❞♦❧✜ ❡ ❚❤❛②❧❛ ❋❡r♥❛♥❞❡s ♣❡❧♦ ❝♦♥st❛♥t❡ ❛♣♦✐♦ q✉❡ ❡❧❡s ❡ ❡❧❛s ♠❡ ❞❡r❛♠ ❡ ❝♦♥t✐♥✉❛♠ ❞❛♥❞♦✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❛ ❯❋▼●✱ ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❝❛♠✐♥❤❛❞❛✱ ▲❡♦♥❛r❞♦ ❈♦♥❡❣✉♥❞❡s✱ ❉✐❧s♦♥ ▲✉❝❛s✱ ❚✐❛❣♦ ❏❛♥✉ár✐♦✱ ❆♠❛❞❡✉ ❆❧♠❡✐❞❛✱ ❱✐t♦r ❆♥❞r❛❞❡✱ ❱✐♥✐❝✐✉s ▼♦r❛✐s✱ ❋r❛♥❦❧✐♥ ❆ss✉♥çã♦✱ ▼❛r❝♦ ❚ú❧✐♦ ❡ ♠✉✐t♦s ♦✉tr♦s✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦ q✉❡ ♣❛s✲ s❛♠♦s ❥✉♥t♦s ❡ ♣♦r t♦❞❛ ❛ ❛❥✉❞❛ q✉❡ ❥á r❡❝❡❜✐ ❞❡❧❡s✳ P♦r ✜♠✱ ♠❛s ♥ã♦ ♠❡♥♦s ✐♠✲ ♣♦rt❛♥t❡✱ ❛❣r❛❞❡ç♦ ♠✉✐t♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❆❧❡①❛♥❞r❡✱ ♣❡❧❛ s✉❛ ♣❛❝✐ê♥❝✐❛✱ ❢r❛♥q✉❡③❛ ❡ ❝❛♠❛r❛❞❛❣❡♠ ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣♦r t❡r ❛❜❡rt♦ ♠✐♥❤❛ ♠❡♥t❡ ❛ ♥♦✈♦s ❝❛♠✐♥❤♦s ❡ ❞❡s❛✜♦s✳

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✏▼② ✇♦r❦ ❝♦♥s✐sts ♦❢ t✇♦ ♣❛rts✿ ♦❢ t❤❡ ♦♥❡ ✇❤✐❝❤ ✐s ❤❡r❡✱ ❛♥❞ ♦❢ ❡✈❡r②t❤✐♥❣ ✇❤✐❝❤ ■ ❤❛✈❡ ♥♦t ✇r✐tt❡♥✳ ❆♥❞ ♣r❡❝✐s❡❧② t❤✐s s❡❝♦♥❞ ♣❛rt ✐s t❤❡ ✐♠♣♦rt❛♥t ♦♥❡✳✑ ✭▲✉❞✇✐❣ ❲✐tt❣❡♥st❡✐♥✮

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❘❡s✉♠♦

❉❛❞♦s ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ k > 2 ❡ ✉♠ ❣r❛❢♦ s✐♠♣❧❡s ♥ã♦✲❞✐r❡❝✐♦♥❛❞♦ G= (V, E)

❝♦♠ ♣❡s♦s r❡❛✐s ♣♦s✐t✐✈♦s ♥❛s ❛r❡st❛s✱ ♦♥❞❡V ❡stá ♣❛rt✐❝✐♦♥❛❞♦ ❡♠k s✉❜❝♦♥❥✉♥t♦s ✭♦✉ ❝❧✉st❡rs✮✱ ♦ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦ ▼í♥✐♠♦ ●❡♥❡r❛❧✐③❛❞♦ ✭P❙❇▼●✮ ❡q✉✐✈❛❧❡ ❛ ❡♥❝♦♥tr❛r ✉♠ s✉❜❣r❛❢♦ ❜✐❝♦♥❡①♦ ❞❡ ❝✉st♦ ♠í♥✐♠♦ S ❞❡ G q✉❡ ❝♦♥t❡♥❤❛ ❡①❛t❛♠❡♥t❡ ✉♠ ✈ért✐❝❡ ❞❡ ❝❛❞❛ ❝❧✉st❡r ❞❡ G✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❢♦r❛♠ ♣r♦♣♦st♦s ✉♠ ❛❧❣♦r✐t♠♦ ❡①❛t♦ ❇r❛♥❝❤✲❛♥❞✲❝✉t ❡ ❤❡✉ríst✐❝❛s ♣❛r❛ s♦❧✉❝✐♦♥❛r ♦ P❙❇▼●✳ ❖ ❛❧❣♦r✐t♠♦ ❇r❛♥❝❤✲❛♥❞✲❝✉t s❡♣❛r❛ q✉❛tr♦ ❝❧❛ss❡s ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s ✈á❧✐❞❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❡♠ q✉❡stã♦✱ t♦❞❛s ❡❧❛s ❣❡♥❡r❛❧✐③❛❞❛s ❞❡ ♦✉tr♦s ♣r♦❜❧❡♠❛s ❞❛ ❧✐t❡r❛t✉r❛ r❡❧❛❝✐♦♥❛❞♦s ❛♦ P❙❇▼●✳ P❛r❛ ❛✈❛❧✐❛r ❛s ❛❜♦r❞❛❣❡♥s ❞❡ s♦❧✉çã♦ ✐♥tr♦❞✉③✐❞❛s✱ ❢♦r❛♠ r❡❛❧✐③❛❞♦s t❡st❡s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❡♠ ✐♥stâ♥❝✐❛s ❞❡ ❜❡♥❝❤♠❛r❦ ❞❡♥s❛s ❡ ❡s♣❛rs❛s✳ P❛r❛ ✉♠ ♥ú♠❡r♦ ❝♦♥s✐❞❡rá✈❡❧ ❞❡❧❛s✱ ❢♦✐ ♣♦ssí✈❡❧ ♣r♦✈❡r ♥♦✈♦s ❝❡rt✐✜❝❛❞♦s ❞❡ ♦t✐♠❛❧✐❞❛❞❡ ❡ ♠❡❤♦r❡s ❧✐♠✐t❡s ✐♥❢❡r✐♦r❡s ❡ s✉♣❡r✐✲ ♦r❡s✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❇✐❝♦♥❡①✐❞❛❞❡ ❡♠ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s✱ Pr♦❥❡t♦ ❞❡ ❘❡❞❡s ❘❡s✐❧✐❡♥t❡s✱

❆❧❣♦r✐t♠♦s ❇r❛♥❝❤✲❛♥❞✲❝✉t✳

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❆❜str❛❝t

●✐✈❡♥ ❛♥ ✐♥t❡❣❡r ♥✉♠❜❡r k > 2 ❛♥❞ ❛♥ ✉♥❞✐r❡❝t❡❞ s✐♠♣❧❡ ❣r❛♣❤ G = (V, E)

✇✐t❤ ♣♦s✐t✐✈❡ r❡❛❧ ✇❡✐❣❤ts ❛ss✐❣♥❡❞ t♦ ✐ts ❡❞❣❡s✱ ✇❤❡r❡ V ✐s ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ k s✉❜s❡ts ✭♦r ❝❧✉st❡rs✮✱ t❤❡ ●❡♥❡r❛❧✐③❡❞ ▼✐♥✐♠✉♠ ❇✐❝♦♥♥❡❝t❡❞ ❙✉❜❣r❛♣❤ Pr♦❜❧❡♠ ✭●▼❇❙P✮ ❛♠♦✉♥ts t♦ ✜♥❞✐♥❣ ❛ ♠✐♥✐♠✉♠✲❝♦st ❜✐❝♦♥♥❡❝t❡❞ s✉❜❣r❛♣❤S ♦❢G✱ s✉❝❤ t❤❛tS❝♦♥t❛✐♥s ❡①❛❝t❧② ♦♥❡ ✈❡rt❡① ❢r♦♠ ❡✈❡r② ❝❧✉st❡r ♦❢ G✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ♣r♦♣♦s❡ ❛♥ ❡①❛❝t ❇r❛♥❝❤✲ ❛♥❞✲❝✉t ❛❧❣♦r✐t❤♠ ❛♥❞ s♦♠❡ ❤❡✉r✐st✐❝s t♦ s♦❧✈❡ P❙❇▼●✳ ❖✉r ❇r❛♥❝❤✲❛♥❞✲❝✉t ❛❧❣♦r✐t❤♠ s❡♣❛r❛t❡s ❢♦✉r ❝❧❛ss❡s ♦❢ ✈❛❧✐❞ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ●▼❇❙P✱ ❛❧❧ ♦❢ t❤❡♠ ❛❞❛♣t❡❞ ❢r♦♠ ✈❛❧✐❞ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ♦t❤❡r ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ ●▼❇❙P✳ ■♥ ♦r❞❡r t♦ ❜❡tt❡r ❛ss❡ss ♦✉r s♦❧✉t✐♦♥ ❛♣♣r♦❛❝❤❡s✱ ✇❡ ❝♦♥❞✉❝t❡❞ s❡✈❡r❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts ✉s✐♥❣ ❞❡♥s❡ ❛♥❞ s♣❛rs❡ ❜❡♥❝❤♠❛r❦ ✐♥st❛♥❝❡s✳ ❋♦r ❛ ❝♦♥s✐❞❡r❛❜❧❡ ❢r❛❝t✐♦♥ ♦❢ t❤♦s❡ ✐♥st❛♥❝❡s✱ ✇❡ ✇❡r❡ ❛❜❧❡ t♦ ♣r♦✈✐❞❡ ♥❡✇ ♦♣t✐♠❛❧✐t② ❝❡rt✐✜❝❛t❡s✱ ❛s ✇❡❧❧ ❛s ❜❡tt❡r ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❜♦✉♥❞s✳

❑❡②✇♦r❞s✿ ❇✐❝♦♥♥❡❝t✐✈✐t② ✐♥ ●r❛♣❤ ❚❤❡♦r②✱ ❙✉r✈✐✈❛❜❧❡ ◆❡t✇♦r❦ ❉❡s✐❣♥ Pr♦❜❧❡♠s✱

❇r❛♥❝❤✲❛♥❞✲❝✉t ❆❧❣♦r✐t❤♠s✳

(16)

(17)

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✹✳✶ ❊✈♦❧✉çã♦ ❞♦s ❣❛♣s ❞❡ ❞✉❛❧✐❞❛❞❡ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡✲ ❣✉♥❞♦s✮ ❞♦ ❛❧❣♦r✐t♠♦ ❇❈ ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❣r✾✻ ❝♦♥s✐❞❡r❛♥❞♦ ❛s q✉❛tr♦ ❝♦♠❜✐♥❛çõ❡s ♣♦ssí✈❡✐s ❞❡ ❝❡♥ár✐♦s ❞❡ s❡♣❛r❛çã♦ ❞❡ ❝♦rt❡s✳ ◆❛ ❧❡❣❡♥❞❛✱ ✏❝s✑✱ ✏♥❝s✑✱ ✏♣♥✑ ❡ ✏❢♣✑ s✐❣♥✐✜❝❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❝✉ts❡ts✱ ♥♦❞❡ ❝✉ts❡ts✱ ❞❡ ♣❛rt✐çã♦ ❞❡ ♥ós ❡ ❞❡ F✲♣❛rt✐çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✷ ❊✈♦❧✉çã♦ ❞♦s ❣❛♣s ❞❡ ❞✉❛❧✐❞❛❞❡ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡❣✉♥✲

❞♦s✮ ❞♦ ❛❧❣♦r✐t♠♦ ❇❈ ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❣r✾✻ ♥♦ ❝❡♥ár✐♦P+_❝ ❝♦♥s✐❞❡r❛♥❞♦ ❛s

três ❝♦♠❜✐♥❛çõ❡s ♣♦ssí✈❡✐s ❞❡ ♣♦❧ít✐❝❛s ❞❡ ✐♥s❡rçã♦ ❞❡ ❝♦rt❡s✱ ❛ s❛❜❡r ✐♥s❡r✐r t♦❞♦s ♦s ❝♦rt❡s ✈✐♦❧❛❞♦s ❡♥❝♦♥tr❛❞♦s✱ ✐♥s❡r✐r ❛♣❡♥❛s ♦ ♠❛✐s ✈✐♦❧❛❞♦ ❡ ✐♥s❡r✐r ❝♦rt❡s θ✲♦rt♦❣♦♥❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✺✳✶ Ps❡✉❞♦✲❝ó❞✐❣♦ ❞❛ ❤❡✉ríst✐❝❛ ❝♦♥str✉t✐✈❛ ✷✲tr❡❡ ♣❛r❛ ♦ P❙❇▼● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

(18)

(19)

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✹✳✶ ◆♦♠❡s✱ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ✭n✮✱ ❞❡ ❛r❡st❛s ✭m✮ ❡ ❞❡ ❝❧✉st❡rs ✭k✮ ❞❛s ✐♥stâ♥❝✐❛s ❞❛ ❚❙P▲■❇ ✉t✐❧✐③❛❞❛s ♥♦s ❡①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷ ▲✐♠✐t❡ ❞❡ P▲✱ ♠❡❧❤♦r ❧✐♠✐t❡ ✐♥❢❡r✐♦r✱ ♠❡❧❤♦r s♦❧✉çã♦ ✐♥t❡✐r❛ ❡ ♥ú♠❡r♦ ❞❡

♥ós ♥❛ ár✈♦r❡ ❞❡ ❡♥✉♠❡r❛çã♦ ♦❜t✐❞♦s ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❇❈ ♥♦s ❝❡♥ár✐♦s P_❝ ❡ P+

❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✹✳✸ ●❛♣s ♥❛ r❛✐③ ❡ ❣❛♣s ❛♦ ✜♥❛❧ ❞❛ ❡①❡❝✉çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❇❈ ♥♦s ❝❡♥ár✐♦s P_❝

❡P+

❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✹✳✹ ❚❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡❣✉♥❞♦s✮ ♣❛r❛ ❛✈❛❧✐❛r ♦ ❧✐♠✐t❡ ❞❡ P▲✱ t❡♠♣♦ t♦✲ t❛❧ ❞❡ ❡①❡❝✉çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❇❈✱ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ♣❛r❛ ♦❜t❡r ❛ ♠❡❧❤♦r s♦❧✉çã♦ ✐♥t❡✐r❛ ❡ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ❛❣r❡❣❛❞♦ ✉s❛❞♦ ♥❛ s❡♣❛r❛çã♦ ❞❛s ❞❡✲ s✐❣✉❛❧❞❛❞❡s ♥♦s ❝❡♥ár✐♦sP_❝ ❡ P+

❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✹✳✺ ❋r❛çõ❡s ♣❡r❝❡♥t✉❛✐s ❞♦s t❡♠♣♦s ❞❡ ❡①❡❝✉çã♦✱ ❡♠ r❡❧❛çã♦ ❛♦ t❡♠♣♦ t♦t❛❧ ❞❡ ❡①❡❝✉çã♦ r❡❧❛t❛❞♦ ♥❛ ❚❛❜❡❧❛ ✹✳✹✱ q✉❡ ♦ ❛❧❣♦r✐t♠♦ ❇❈ ✉s❛ ♣❛r❛ ❛✈❛❧✐❛r ♦ ❧✐♠✐t❡ ❞❡ P▲ ❡ ♣❛r❛ s❡♣❛r❛r ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ♥♦s ❝❡♥ár✐♦s P_❝ ❡P+

❝✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✹✳✻ ●❛♣s ♥❛ r❛✐③ ❡ ♦s t❡♠♣♦s ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡❣✉♥❞♦s✮ ❞♦ ♥♦ss♦ ❛❧❣♦r✐t♠♦ ❇❈ ♣❛r❛ ❛✈❛❧✐❛r ♦ ❧✐♠✐t❡ ❞❡ P▲ ❝♦♠ ❛ ❢♦r♠✉❧❛çã♦ P+

❝ ❡ ❞♦ ❛❧❣♦r✐t♠♦ ❇❈ ♣r♦✲

♣♦st♦ ♣♦r ❑❡r✐✈✐♥ ❡t ❛❧✳ ❬✷✵✵✹❪ ♣❛r❛ ♦ P❙❇▼✳ P❛r❛ ✉♠❛ ❝♦♠♣❛r❛çã♦ ❥✉st❛ ❞♦s t❡♠♣♦s ❞❡ ❡①❡❝✉çã♦✱ ❡♠ ❞❡❝♦rrê♥❝✐❛ ❞❛s ❞✐❢❡r❡♥ç❛s ❞♦s ❛♠❜✐❡♥t❡s ❝♦♠✲ ♣✉t❛❝✐♦♥❛✐s ❞❡ ❛♠❜♦s ♦s tr❛❜❛❧❤♦s✱ ❝❛❜❡ r❡ss❛❧t❛r q✉❡ ♦s ♥♦ss♦s r❡s✉❧t❛❞♦s ♣r❡❝✐s❛♠ s❡r ♠✉❧t✐♣❧✐❝❛❞♦s ♣♦r ✉♠ ❢❛t♦r ❞❡8✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✹✳✼ ●❛♣s ❛♦ ✜♥❛❧ ❞❛ ❡①❡❝✉çã♦ ❡ t❡♠♣♦ t♦t❛❧ ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡❣✉♥❞♦s✮ ❞♦ ❛❧❣♦✲ r✐t♠♦ ❇❈ ❝♦♠ ❛ ❢♦r♠✉❧❛çã♦P+

❝ ❝♦♥s✐❞❡r❛♥❞♦ ❞✐❢❡r❡♥t❡s ♥í✈❡✐s ❞❡ ❞❡♥s✐❞❛❞❡

❞❡ ❛r❡st❛s ♥❛s ✐♥stâ♥❝✐❛s ❞❡ t❡st❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✽ ●❛♣s ❛♦ ✜♥❛❧ ❞❛ ❡①❡❝✉çã♦ ❡ t❡♠♣♦ t♦t❛❧ ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡❣✉♥❞♦s✮ ❞♦ ❛❧✲

❣♦r✐t♠♦ ❇❈ ❝♦♠ ❛ ❢♦r♠✉❧❛çã♦P+_❝ ❝♦♥s✐❞❡r❛♥❞♦ ❞✐❢❡r❡♥t❡s ♥í✈❡✐s ❞❡ ❛❣❧♦♠✲

❡r❛çã♦ ❞♦s ✈ért✐❝❡s ♥❛s ✐♥stâ♥❝✐❛s ❞❡ t❡st❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

(20)

✹✳✾ ◆ú♠❡r♦ ❞❡ ❝♦rt❡s ❡♥❝♦♥tr❛❞♦s ❞❛s ❝❧❛ss❡s ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s s❡♣❛r❛❞❛s ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❇❈ ❝♦♥s✐❞❡r❛♥❞♦ ❞♦✐s ❝❡♥ár✐♦s ❞❡ ❡①❡❝✉çã♦ ❞✐st✐♥t♦s P_❝ ❡ P+

❝✳ ✳ ✳ ✺✶

✺✳✶ ❈✉st♦s ❞❛s s♦❧✉çõ❡s ❡♥❝♦♥tr❛❞❛s ♣❡❧❛s ❤❡✉ríst✐❝❛s ❝♦♥str✉t✐✈❛s ✷✲tr❡❡ ✭✷❚✮ ❡ ❝♦✈❡r✲❛♥❞✲st✐t❝❤ ✭❈❙✮ q✉❛♥❞♦ ✉s❛❞❛s ✐s♦❧❛❞❛♠❡♥t❡✱ q✉❛♥❞♦ ❝♦♠❜✐♥❛❞❛s ❝♦♠ ❛ ❤❡✉ríst✐❝❛ ❞❡ r❡✜♥❛♠❡♥t♦ ❞❡str♦②✲❛♥❞✲r❡st♦r❡ ✭❉❘✮ ❡ ❝♦♠ ❛s ❜✉s❝❛s ❧♦❝❛✐s ✭❱◆❉✮✳ ❈♦♠♣❛r❛♠♦s t❛♠❜é♠ ♦s r❡s✉❧t❛❞♦s ❝♦♠ ♦ ❝✉st♦ ❞❛ ♣r✐♠❡✐r❛ s♦❧✉çã♦ ✐♥t❡✐r❛ ❡♥❝♦♥tr❛❞❛ ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❇❈ ❝♦♠ ❛ ❢♦r♠✉❧❛çã♦P+

❝ ❛♦ ❧♦♥❣♦

❞❛ ár✈♦r❡ ❞❡ ❡♥✉♠❡r❛çã♦ q✉❛♥❞♦ ♥ã♦ é ❢♦r♥❡❝✐❞❛ ❛ ❡❧❡ ♥❡♥❤✉♠❛ s♦❧✉çã♦ ✐♥✐❝✐❛❧✳ ✻✾ ✺✳✷ ❚❡♠♣♦s ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡❣✉♥❞♦s✮ ❞❛s ❤❡✉ríst✐❝❛s ❝♦♥str✉t✐✈❛s ✷✲tr❡❡ ✭✷❚✮

❡ ❝♦✈❡r✲❛♥❞✲st✐t❝❤ ✭❈❙✮ ♣❛r❛ ❛ ❡♥❝♦♥tr❛r ❛ s♦❧✉çã♦ ✐♥✐❝✐❛❧ q✉❛♥❞♦ ✉s❛❞❛s ✐s♦❧❛❞❛♠❡♥t❡✱ q✉❛♥❞♦ ❝♦♠❜✐♥❛❞❛s ❝♦♠ ❛ ❤❡✉ríst✐❝❛ ❞❡ r❡✜♥❛♠❡♥t♦ ❞❡str♦②✲ ❛♥❞✲r❡st♦r❡ ✭❉❘✮ ❡ ❝♦♠ ❛s ❜✉s❝❛s ❧♦❝❛✐s ✭❱◆❉✮✳ ❈♦♠♣❛r❛♠♦s t❛♠❜é♠ ♦s r❡s✉❧t❛❞♦s ❝♦♠ ♦ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ❣❛st♦ ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❇❈ ❝♦♠ ❛ ❢♦r♠✉✲ ❧❛çã♦ P+_❝ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ♣r✐♠❡✐r❛ s♦❧✉çã♦ ✐♥t❡✐r❛ ❛♦ ❧♦♥❣♦ ❞❛ ár✈♦r❡ ❞❡

❡♥✉♠❡r❛çã♦✱ ✉s❛♥❞♦ ✉♠❛ ❡str❛té❣✐❛ ❞❡ ❝❛♠✐♥❤❛♠❡♥t♦ ❡♠ ❧❛r❣✉r❛✱ q✉❛♥❞♦ ♥ã♦ é ❢♦r♥❡❝✐❞❛ ❛ ❡❧❡ ♥❡♥❤✉♠❛ s♦❧✉çã♦ ✐♥✐❝✐❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✺✳✸ ❚❡♠♣♦ t♦t❛❧ ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡❣✉♥❞♦s✮ ❞♦ ❛❧❣♦r✐t♠♦ ❇❈ ❝♦♠ ❛ ❢♦r♠✉❧❛çã♦

P+

❝✱ t❡♠♣♦ t♦t❛❧ ❞❡ ❡①❡❝✉çã♦ ❞♦ ♣r♦❝❡❞✐♠❡♥t♦s ❞❡ ❜✉s❝❛ ❧♦❝❛❧ ✭❱◆❉✮ ❞✉✲

r❛♥t❡ ❛ ❡①❡❝✉çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❇❈ ❡ s✉❛ ❢r❛çã♦ ♣❡r❝❡♥t✉❛❧ ❡♠ r❡❧❛çã♦ ❛♦ t❡♠♣♦ t♦t❛❧ ❞❡ ❡①❡❝✉çã♦ ❞♦ ❇❈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✺✳✹ ❈✉st♦s ❞❛s ♠❡❧❤♦r❡s s♦❧✉çõ❡s ✐♥t❡✐r❛s ♦❜t✐❞❛s ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❇❈ ❝♦♠ ❛ ❢♦r✲

♠✉❧❛çã♦P+_❝ ❡ ❝✉st♦s ❞❛s ♠❡❧❤♦r❡s s♦❧✉çõ❡s q✉❡ ❢♦r❛♠ ♦❜t✐❞❛s ♣❡❧❛ ❤❡✉ríst✐❝❛

❝♦✈❡r✲❛♥❞✲st✐t❝❤ ✭❈❙✮ ✐♥t❡❣r❛❞❛ ❝♦♠ ❛s ❜✉s❝❛s ❧♦❝❛✐s ✭❱◆❉✮✳ ▼♦str❛♠♦s t❛♠❜é♠ ♦s ❣❛♣s ❞❡ss❛s s♦❧✉çõ❡s ❤❡✉ríst✐❝❛s ❝❛❧❝✉❧❛❞♦s ❡♠ r❡❧❛çã♦ ❛♦s ❝✉s✲ t♦s ❞❛s s♦❧✉çõ❡s ót✐♠❛s ♦❜t✐❞❛s ♣❡❧♦ ♦ ❛❧❣♦r✐t♠♦ ❇❈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✺✳✺ ❚❡♠♣♦s ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡❣✉♥❞♦s✮ ❡ ❝✉st♦s ❞❛s s♦❧✉çõ❡s ❡♥❝♦♥tr❛❞❛s ♣❡❧❛

❝♦♠❜✐♥❛çã♦ ❞❛ ❤❡✉ríst✐❝❛ ❝♦♥str✉t✐✈❛ ❝♦✈❡r✲❛♥❞✲st✐t❝❤ ✭❈❙✮ ❝♦♠ ❛s ❜✉s❝❛s ❧♦❝❛✐s ✭❱◆❉✮ ❡ ❞❛s s♦❧✉çõ❡s ❡♥❝♦♥tr❛❞❛s ♣❡❧♦ ❛❧❣♦r✐t♠♦ ♠❡♠ét✐❝♦ ♣r♦♣♦st♦ ♣♦r ❍✉ ✫ ❘❛✐❞❧ ❬✷✵✵✾❪✳ P❛r❛ ✉♠❛ ❝♦♠♣❛r❛çã♦ ❥✉st❛ ❞♦s r❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✼✸ ✺✳✻ ❚♦t❛❧ ❞❡ ❡①❡❝✉çõ❡s ❞❛s ❜✉s❝❛s ❧♦❝❛✐s ❞✉r❛♥t❡ ❛ ❡①❡❝✉çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❇❈

❝♦♠ ❛ ❢♦r♠✉❧❛çã♦ P+_❝✱ ♥ú♠❡r♦ ❞❡ ❡①❡❝✉çõ❡s ♣♦s✐t✐✈❛s✱ ✐✳❡✳✱ ❡①❡❝✉çõ❡s q✉❡

❡❢❡t✐✈❛♠❡♥t❡ ❛♣r✐♠♦r❛r❛♠ ❛ ♠❡❧❤♦r s♦❧✉çã♦ ✐♥t❡✐r❛ ❝♦rr❡♥t❡✱ ❛s ❢r❛çõ❡s ♣❡r✲ ❝❡♥t✉❛✐s ❞❛s ❡①❡❝✉çõ❡s ♣♦s✐t✐✈❛s ❡♠ r❡❧❛çã♦ ❛♦ t♦t❛❧ ❞❡ ❡①❡❝✉çõ❡s ❡ ❛ ❛✉t♦r✐❛ ❞❛ ♠❡❧❤♦r s♦❧✉çã♦ ✐♥t❡✐r❛ r❡❧❛t❛❞❛ ♥❛ ❚❛❜❡❧❛ ✹✳✷✱ ✐✳❡✳✱ ♦ ♠ét♦❞♦ q✉❡ ♦❜t❡✈❡ ❛ ♠❡❧❤♦r s♦❧✉çã♦ ✐♥t❡✐r❛ ❝♦♥❤❡❝✐❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹

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❙✉♠ár✐♦

❆❣r❛❞❡❝✐♠❡♥t♦s ✐①

❘❡s✉♠♦ ①✐✐✐

❆❜str❛❝t ①✈

▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✈✐✐

▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✐①

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ Pr✐♥❝✐♣❛✐s ❈♦♥tr✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❖r❣❛♥✐③❛çã♦ ❞❛ ❉✐ss❡rt❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✷ ❖ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦ ▼í♥✐♠♦ ●❡♥❡r❛❧✐③❛❞♦ ✺

✷✳✶ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❉❡✜♥✐çã♦ ❞♦ Pr♦❜❧❡♠❛ ❡ Pr♦❜❧❡♠❛s ❘❡❧❛❝✐♦♥❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✸ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸✳✶ ❆❧❣♦r✐t♠♦s ❞❡ ❆♣r♦①✐♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸✳✷ ❍❡✉ríst✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸✳✸ ❋♦r♠✉❧❛çõ❡s ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ❡ ❆❧❣♦r✐t♠♦s ❊①❛t♦s ✳ ✳ ✳ ✳ ✶✷ ✷✳✹ ❖ P❙❇▼● ❡ ❙✉❛s Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✸ ❋♦r♠✉❧❛çõ❡s ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ♣❛r❛ ♦ P❙❇▼● ✶✾

✸✳✶ ❋♦r♠✉❧❛çã♦ ◆ã♦✲❉✐r❡❝✐♦♥❛❞❛ ❇❛s❡❛❞❛ ❡♠ ❝✉ts❡ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✶✳✶ ◆♦✈❛s ❉❡s✐❣✉❛❧❞❛❞❡s ❱á❧✐❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✷ ❋♦r♠✉❧❛çã♦ ❉✐r❡❝✐♦♥❛❞❛ ❇❛s❡❛❞❛ ❡♠ ❞✐r❡❝t❡❞ ❝✉ts❡ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✸ ❋♦r♠✉❧❛çã♦ ❉✐r❡❝✐♦♥❛❞❛ ❇❛s❡❛❞❛ ❡♠ ❋❧✉①♦s ▼✉❧t✐✲♣r♦❞✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✹ ❯♠ ▼ét♦❞♦ ❞❡ ❙♦❧✉çã♦ ❊①❛t❛ ♣❛r❛ ♦ P❙❇▼● ✷✼

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✹✳✶ ❆❧❣♦r✐t♠♦ ❇r❛♥❝❤✲❛♥❞✲❝✉t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✶✳✶ Pr♦❝❡❞✐♠❡♥t♦s ❞❡ ❙❡♣❛r❛çã♦ ❞❡ ❉❡s✐❣✉❛❧❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✳✶✳✷ ❉❡t❛❧❤❡s ❞❡ ■♠♣❧❡♠❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✶✳✸ ❊①♣❡r✐♠❡♥t♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✺ ▼ét♦❞♦s ❞❡ ❙♦❧✉çã♦ ❍❡✉ríst✐❝❛ ♣❛r❛ ♦ P❙❇▼● ✺✸

✺✳✶ ❍❡✉ríst✐❝❛s ❈♦♥str✉t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✶✳✶ ❍❡✉ríst✐❝❛ ✷✲tr❡❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✺✳✶✳✷ ❍❡✉ríst✐❝❛ ❝♦✈❡r✲❛♥❞✲st✐t❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✺✳✷ ❍❡✉ríst✐❝❛ ❞❡ ❘❡✜♥❛♠❡♥t♦ ❞❡str♦②✲❛♥❞✲r❡st♦r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺✳✸ Pr♦❝❡❞✐♠❡♥t♦s ❞❡ ❇✉s❝❛ ▲♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✺✳✸✳✶ ❱✐③✐♥❤❛♥ç❛ ❝❧✉st❡r✲❤❡❛❞ ❡①❝❤❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✺✳✸✳✷ ❱✐③✐♥❤❛♥ç❛ ❡❞❣❡ s✇❛♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✺✳✹ ❊①♣❡r✐♠❡♥t♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

✻ ❈♦♥❝❧✉sõ❡s ❡ ❚r❛❜❛❧❤♦s ❋✉t✉r♦s ✼✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✾

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❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

◆♦ ❞✐❛ ✽ ❞❡ ♠❛✐♦ ❞❡ ✶✾✾✶✱ ♦ ❲❛❧❧ ❙tr❡❡t ❏♦✉r♥❛❧ ❬❙t♦❡r✱ ✶✾✾✷❪ ♥♦t✐❝✐♦✉ q✉❡ ♦ ❛❡r♦♣♦rt♦ ✐♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ❖✬❍❛r❡✱ ❡♠ ❈❤✐❝❛❣♦✱ ♥♦s ❊st❛❞♦s ❯♥✐❞♦s✱ ❤❛✈✐❛ ❡♥tr❛❞♦ ❡♠ ❝♦❧❛♣s♦✳ ▲✐♥❤❛s ❞❡ ❡♠❡r❣ê♥❝✐❛ ✾✶✶ ✜❝❛r❛♠ ✐♥❛❧❝❛♥çá✈❡✐s ❡ ❝❛✐①❛s ❡❧❡trô♥✐❝♦s ✐♥♦✲ ♣❡r❛♥t❡s✳ ❖ ♠♦t✐✈♦ ❞♦ ❛❝✐❞❡♥t❡ ❢♦✐ ✉♠ ✐♥❝ê♥❞✐♦ q✉❡ ❞❛♥✐✜❝♦✉ ♦ ♠❛q✉✐♥ár✐♦ ❞❡ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❡♠♣r❡s❛ ❞❡ t❡❧❡❝♦♠✉♥✐❝❛çõ❡s✱ ♦❝❛s✐♦♥❛♥❞♦ ❛ ❞❡s❝♦♥❡①ã♦ ❞❛ s✉❛ r❡❞❡ ❞❡ t❡❧❡❢♦♥✐❛✳ ❊♣✐só❞✐♦s ❝♦♠♦ ❡ss❡ ♥ã♦ sã♦ r❛r♦s✳ ❯♠❛ ❧✐çã♦ q✉❡ ❡❧❡s ♥♦s ❡♥s✐♥❛♠ é ❛ ❞❡ q✉❡ r❡❞❡s ❞❡ ❝♦♠✉♥✐❝❛çã♦ ♣r❡❝✐s❛♠ s❡r ♣❧❛♥❡❥❛❞❛s ❡ ❝♦♥str✉í❞❛s ❞❡ ♠♦❞♦ ❛ ♣♦ss✉✐r r❡s✐❧✐ê♥❝✐❛✳

❉✐③❡♠♦s q✉❡ ✉♠❛ r❡❞❡ ❞❡ ❝♦♠✉♥✐❝❛çã♦ é ❝♦♥s✐❞❡r❛❞❛ r❡s✐❧✐❡♥t❡ s❡ ❡❧❛ é ❝❛♣❛③ ❞❡ s❡ ♠❛♥t❡r ❡♠ ❢✉♥❝✐♦♥❛♠❡♥t♦ ♠❡s♠♦ ❛♣ós ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❢❛❧❤❛s ❡♠ s❡✉s ❝♦♠♣♦♥❡♥t❡s ✐♥t❡r♥♦s✳ ❆ t❛r❡❢❛ ❞❡ s❡ ♣r♦❥❡t❛r ✉♠❛ r❡❞❡ r❡s✐❧✐❡♥t❡ é ❝♦♠♣❧✐❝❛❞❛ ❡ ❡♥❣❧♦❜❛ ♠ú❧t✐♣❧❛s ❢❛s❡s✱ ✉♠❛ ❞❡❧❛s✱ ❡♥❢❛t✐③❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦✱ s❡♥❞♦ ♦ ❝❤❛♠❛❞♦ ❞❡s❡♥❤♦ t♦♣♦❧ó❣✐❝♦ ❞❛ r❡❞❡✳ ◆❡ss❛ ❡t❛♣❛✱ ♦ ♦❜❥❡t✐✈♦ é ❞❡❝✐❞✐r ♦♥❞❡ ♦s ♥ós ❞❛ r❡❞❡ s❡rã♦ ♣♦s✐❝✐♦♥❛❞♦s ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❝♦♠♦ ❡❧❡s s❡rã♦ ❝♦♥❡❝t❛❞♦s ✉♥s ❛♦s ♦✉tr♦s✳ P❛r❛ t❛♥t♦✱ ❞❡✈❡✲s❡ ❧❡✈❛r ❡♠ ❝♦♥t❛✱ ❡♥tr❡ ♦✉tr♦s ❢❛t♦r❡s✱ ♦s ❝✉st♦s ❞❡ ✐♥st❛❧❛çã♦ ❡ ❞❡ ✐♥t❡r❝♦♠✉♥✐❝❛çã♦ ❞♦s ♥ós✳ ◆♦ ❝♦♥t❡①t♦ ❞❡ r❡❞❡s ❞❡ ❝♦♠♣✉t❛❞♦r❡s✱ ❝♦♠ ❛ ♣♦♣✉❧❛r✐③❛çã♦ ❞❛ t❡❝♥♦❧♦❣✐❛ ❞❡ ✜❜r❛ ót✐❝❛ ❞❡✈✐❞♦ ❛ s✉❛ ❛❧t❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❡ ❧❛r❣✉r❛ ❞❡ ❜❛♥❞❛✱ ♠ét♦❞♦s ❛t✉❛✐s ❞❡ ♣r♦❥❡t♦ ❞❡ r❡❞❡s ❝♦st✉♠❛♠ ❢❛✈♦r❡❝❡r t♦♣♦❧♦❣✐❛s ♠❛✐s ❡s♣❛rs❛s✳ ❚❛❧ ❞❡❝✐sã♦ ❛✉♠❡♥t❛ ♦ ♣♦t❡♥❝✐❛❧ r✐s❝♦ ❞❡ ✐♥t❡rr✉♣çã♦ ❞♦s s❡r✈✐ç♦s ♣r♦✈✐❞♦s ♥❛ r❡❞❡ ❡♠ ❞❡❝♦rrê♥❝✐❛ ❞❡ ❛❝✐❞❡♥t❡s ♥❛t✉r❛✐s ♦✉ ❞❛♥♦s ❢ís✐❝♦s✳ P♦r ❡ss❛ r❛③ã♦✱ ♦ ♣r♦❥❡t♦ ❞❡ r❡❞❡s r❡s✐❧✐❡♥t❡s ❣❛♥❤❛ ✉♠ ♣❛♣❡❧ ❝r✉❝✐❛❧✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s✱ ❛ q✉❡stã♦ ❞❛ r❡s✐❧✐ê♥❝✐❛ ❡♠ r❡❞❡s ❞❡ ❝♦♠✲ ♣✉t❛❞♦r❡s s❡ ❡s♣❡❧❤❛ ♥♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥❡①✐❞❛❞❡✳ ❯♠❛ ❢♦r♠❛ ❞❡ ❛t✐♥❣✐r r❡s✐❧✐ê♥❝✐❛✱ ❛♦ s❡ ♣r♦❥❡t❛r ❛ t♦♣♦❧♦❣✐❛ ❞❡ ✉♠❛ r❡❞❡✱ é ❣❛r❛♥t✐r q✉❡ ❡①✐st❛♠ ♠ú❧t✐♣❧♦s ❝❛♠✐♥❤♦s ❞✐s❥✉♥t♦s ❡♥tr❡ q✉❛❧q✉❡r ♣❛r ❞❡ ♥ós ❞❛ r❡❞❡✳ ❊st✉❞♦s ❥á ❝♦♥✜r♠❛r❛♠ q✉❡ t♦♣♦❧♦❣✐❛s ❜✐❝♦♥❡①❛s ❝♦♥s❡❣✉❡♠✱ ❝♦♠ ❜❛✐①♦ ❝✉st♦✱ ♣r♦✈❡r ✉♠ ♥í✈❡❧ s❡❣✉r♦ ❞❡ r❡s✐❧✐ê♥❝✐❛✳ ❋♦r♠❛❧♠❡♥t❡✱ ✉♠

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✷ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦

❣r❛❢♦ é ❝❤❛♠❛❞♦ ❞❡ ✈ért✐❝❡✲❜✐❝♦♥❡①♦✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❜✐❝♦♥❡①♦✱ s❡ ❡①✐st❡♠✱ ♥♦ ♠í♥✐♠♦✱ ❞♦✐s ❝❛♠✐♥❤♦s ✈ért✐❝❡✲❞✐s❥✉♥t♦s ❡♥tr❡ q✉❛✐sq✉❡r ❞♦✐s ✈ért✐❝❡s ❞✐st✐♥t♦s ♥❡ss❡ ❣r❛❢♦✳

◆♦ â♠❜✐t♦ ❞❛ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛✱ ♦ ♣r♦❜❧❡♠❛ ❞♦ ♣r♦❥❡t♦ ❞❡ r❡❞❡s r❡✲ s✐❧✐❡♥t❡s s❡ ❡♥❝❛✐①❛ ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠❛✐s ❛♠♣❧❛ ❞❡ ♣r♦❜❧❡♠❛s ❝♦♥❤❡❝✐❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ❛❝❛❞ê♠✐❝❛ ❝♦♠♦ ♣r♦❜❧❡♠❛s ❞❡ ♣r♦❥❡t♦ ❞❡ r❡❞❡s✳ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ r❡str✐çõ❡s ❞❡ ❝♦♥❡①✐❞❛❞❡ ❡ ♣❡s♦s ❛ss♦❝✐❛❞♦s ❛♦s ✈ért✐❝❡s ❡ ❛r❡st❛s ❞❡ ✉♠ ❣r❛❢♦ G✱ ♦ Pr♦❜❧❡♠❛ ❞♦ Pr♦❥❡t♦ ❞❡ ❘❡❞❡s ✭PP❘✮ ❜✉s❝❛ ❛ ♦❜t❡♥çã♦ ❞❡ ✉♠ s✉❜❣r❛❢♦ ❞❡ G q✉❡ ❝✉♠♣r❛ ❛s r❡s✲ tr✐çõ❡s ✐♠♣♦st❛s ❡ q✉❡ ♣♦ss✉❛ ❝✉st♦ ♠í♥✐♠♦✱ ✐✳❡✳✱ t❛❧ q✉❡ ❛ s♦♠❛ ❞♦s ♣❡s♦s ❞♦s ✈ért✐❝❡s ❡ ❛r❡st❛s ♣r❡s❡♥t❡s ♥❛ s♦❧✉çã♦ s❡❥❛ ♠í♥✐♠❛✳

❖ PP❘ t❡♠ r❡❝❡❜✐❞♦ ❜❛st❛♥t❡ ❛t❡♥çã♦ ♥♦s ♠❡✐♦s ❛❝❛❞ê♠✐❝♦ ❡ ✐♥❞✉str✐❛❧ r❡❝❡♥t❡✲ ♠❡♥t❡✳ ❊♠ ♣❛rt❡✱ ✐ss♦ s❡ ❥✉st✐✜❝❛ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ✐♥ú♠❡r♦s ♣r♦❜❧❡♠❛s ♠✉✐t♦ ❡st✉❞❛❞♦s ❡♠ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛✱ ❡✳❣✳✱ ♦ Pr♦❜❧❡♠❛ ❞♦ ❈❛♠✐♥❤♦ ▼í♥✐♠♦✱ ♦ Pr♦❜❧❡♠❛ ❞❛ ➪r✈♦r❡ ●❡r❛❞♦r❛ ▼í♥✐♠❛✱ ❡♥tr❡ ♦✉tr♦s✱ ♣♦❞❡♠ s❡r ❢❛❝✐❧♠❡♥t❡ r❡❞✉③✐❞♦s ❛ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ PP❘✱ ♦ q✉❡ ❞❡♠♦♥str❛ s✉❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❣❡♥❡r❛❧✐③❛çã♦✳ ❯♠❛ ❡①t❡♥sã♦ r❡❧❡✈❛♥t❡ ❞♦ PP❘ é ♦ ❝❤❛♠❛❞♦ Pr♦❜❧❡♠❛ ❞♦ Pr♦❥❡t♦ ❞❡ ❘❡❞❡s ●❡♥❡r❛❧✐③❛❞♦ ✭PP❘●✮✱ q✉❡ s❡ ❞✐❢❡r❡♥❝✐❛ ❞♦ PP❘ ♥♦ ❛s♣❡❝t♦ ❞❡ q✉❡ s✉❛s r❡str✐çõ❡s sã♦ ❡①♣r❡ss❛s ♥ã♦ ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s ✈ért✐❝❡s ❞♦ ❣r❛❢♦ q✉❡ ♠♦❞❡❧❛ ❛ r❡❞❡✱ ♠❛s ♣❛r❛ ❛❣❧♦♠❡r❛❞♦s ❞❡ ✈ért✐❝❡s ✭♦✉ ❝❧✉st❡rs✮✳ ❉✐✈❡rs♦s ♦✉tr♦s ♣r♦❜❧❡♠❛s ❞✐t♦s ✏❣❡♥❡r❛❧✐③❛❞♦s✑✱ s❡♠❡❧❤❛♥t❡s ❛♦ PP❘●✱ ❥á ❢♦r❛♠ ❡st✉❞❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛✱ ❡♥tr❡ ❡❧❡s ❝✐t❛♠♦s ♦ Pr♦❜❧❡♠❛ ❞❛ ➪r✈♦r❡ ●❡r❛❞♦r❛ ▼í♥✐♠❛ ●❡♥❡r❛❧✐③❛❞♦ ❬▼②✉♥❣ ❡t ❛❧✳✱ ✶✾✾✺❪ ❡ ♦ Pr♦❜❧❡♠❛ ❞♦ ❈❛✐①❡✐r♦ ❱✐❛❥❛♥t❡ ●❡♥❡r✲ ❛❧✐③❛❞♦ ❬❆♣♣❧❡❣❛t❡ ❡t ❛❧✳✱ ✷✵✵✻❪✳

◆❡st❛ ❞✐ss❡rt❛çã♦✱ ❛❜♦r❞❛♠♦s ✉♠❛ ❡s♣❡❝✐❛❧✐③❛çã♦ ◆P✲❞✐❢í❝✐❧ ❞♦ PP❘✱ ❞❡♥♦♠✐♥❛❞❛ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦ ▼í♥✐♠♦ ●❡♥❡r❛❧✐③❛❞♦ ✭P❙❇▼●✮✳ ❙❡♥❞♦ G ✉♠ ❣r❛❢♦ ❝♦♠ ♣❡s♦s r❡❛✐s ♣♦s✐t✐✈♦s ♥❛s ❛r❡st❛s ❡ ❝♦♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s V ♣❛rt✐❝✐♦♥❛❞♦ ❡♠ k ❝❧✉st❡rs✱ ♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦ k > 2✱ ♦ P❙❇▼● ❡q✉✐✈❛❧❡ ❛ ❡♥❝♦♥tr❛r ✉♠ s✉❜❣r❛❢♦

❜✐❝♦♥❡①♦ ❞❡G ❞❡ ❝✉st♦ ♠í♥✐♠♦ ❝♦♥t❡♥❞♦ ❡①❛t❛♠❡♥t❡ ✉♠ ✈ért✐❝❡ ❞❡ ❝❛❞❛ ❝❧✉st❡r✳ ❆♦ ❝♦♥trár✐♦ ❞❛ s✉❛ ✈❡rsã♦ ♥ã♦✲❣❡♥❡r❛❧✐③❛❞❛✱ ✐✳❡✳✱ ❞♦ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦ ▼í✲ ♥✐♠♦ ✭P❙❇▼✮✱ ♦ P❙❇▼● ❢♦✐ ♣♦✉❝♦ ❡st✉❞❛❞♦✳ ❖ ♣r✐♠❡✐r♦ tr❛❜❛❧❤♦ q✉❡ ♣r♦♣ôs ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞❡ s♦❧✉çã♦ ♣❛r❛ ♦ P❙❇▼● ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❍✉ ✫ ❘❛✐❞❧ ❬✷✵✵✾❪✳ ❉❡s❞❡ ❡♥tã♦✱ ♣❡❧♦ ♦ q✉❡ s❡ t❡♠ ❝♦♥❤❡❝✐♠❡♥t♦✱ s♦♠❡♥t❡ ♠❛✐s ✉♠ tr❛❜❛❧❤♦✱ ❝♦♥❞✉③✐❞♦ ♣♦r P❛❣❛❝③ ❡t ❛❧✳ ❬✷✵✶✵❪✱ ❡①♣❧♦r♦✉ ♦ P❙❇▼●✳

✶✳✶ Pr✐♥❝✐♣❛✐s ❈♦♥tr✐❜✉✐çõ❡s

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✶✳✷✳ ❖r❣❛♥✐③❛çã♦ ❞❛ ❉✐ss❡rt❛çã♦ ✸

• ❉✐s❝✉ssã♦ ❞❡ três ❢♦r♠✉❧❛çõ❡s ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ♣❛r❛ ♦ P❙❇▼●✳ ❉✉❛s ❢♦r✲ ♠✉❧❛çõ❡s✱ ✉♠❛ ❞✐r❡❝✐♦♥❛❞❛ ❡ ❛ ♦✉tr❛ ♥ã♦✲❞✐r❡❝✐♦♥❛❞❛✱ s❡ ❜❛s❡✐❛♠ ❡♠ ✉♠ ♥ú♠❡r♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ r❡str✐çõ❡s ❝✉ts❡ts ❡ ❡♠ r❡s✉❧t❛❞♦s ❞❡ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s q✉❡ ❣❛r❛♥✲ t❡♠ s✉❛ ✈❛❧✐❞❛❞❡✳ ❆ t❡r❝❡✐r❛ ❢♦r♠✉❧❛çã♦ ✉s❛ ✉♠ ♥ú♠❡r♦ ♣♦❧✐♥♦♠✐❛❧✱ ♥❛ q✉❛♥t✐✲ ❞❛❞❡ ❞❡ ✈ért✐❝❡s ❡ ❛r❡st❛s ❞❛ ✐♥stâ♥❝✐❛✱ ❞❡ r❡str✐çõ❡s ❞❡ ✢✉①♦s ♠✉❧t✐✲♣r♦❞✉t♦✳

• ❆♣r❡s❡♥t❛çã♦ ❞❡ ❞✉❛s ❝❧❛ss❡s ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s ✈á❧✐❞❛s ♣❛r❛ ♦ ♣♦❧✐t♦♣♦ ❞❛s s♦❧✉çõ❡s ✐♥t❡✐r❛s ❞♦ P❙❇▼●✱ ❛❞❛♣t❛❞❛s ❞❡ ♦✉tr❛s ❞❡s✐❣✉❛❧❞❛❞❡s ✈á❧✐❞❛s ♣❛r❛ ♦ P❙❇▼✳

• ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ❡①❛t♦ ❇r❛♥❝❤✲❛♥❞✲❝✉t✱ ❞✉❛s ❤❡✉ríst✐❝❛s ❝♦♥s✲ tr✉t✐✈❛s✱ ✉♠❛ ❤❡✉ríst✐❝❛ ❞❡ r❡✜♥❛♠❡♥t♦ ❡ ❞✉❛s ❡str✉t✉r❛s ❞❡ ✈✐③✐♥❤❛♥ç❛ ❞❡ ❜✉s❝❛ ❧♦❝❛❧ ♣❛r❛ ♦ P❙❇▼●✳

• ◆♦✈♦s ❝❡rt✐✜❝❛❞♦s ❞❡ ♦t✐♠❛❧✐❞❛❞❡✱ ♠❡❧❤♦r❡s ❧✐♠✐t❡s ✐♥❢❡r✐♦r❡s ❡ s✉♣❡r✐♦r❡s ♣❛r❛ ✐♥stâ♥❝✐❛s ❞♦ P❙❇▼● ❞❡ ♣❡q✉❡♥♦✱ ♠é❞✐♦ ❡ ❣r❛♥❞❡ ♣♦rt❡ t❛♠❜é♠ ✉s❛❞❛s ❡♠ ♦✉tr♦s tr❛❜❛❧❤♦s ❞❛ ❧✐t❡r❛t✉r❛✳

✶✳✷ ❖r❣❛♥✐③❛çã♦ ❞❛ ❉✐ss❡rt❛çã♦

❊st❛ ❞✐ss❡rt❛çã♦ ❡stá ❡str✉t✉r❛❞❛ ❡♠ s❡✐s ❝❛♣ít✉❧♦s✳ ❖ r❡st❛♥t❡ ❞♦ t❡①t♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

❈❛♣ít✉❧♦ ✷✳ ❬❖ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦ ▼í♥✐♠♦ ●❡♥❡r❛❧✐③❛❞♦❪ ❖ P❙❇▼● é ❞❡✜♥✐❞♦ ❢♦r♠❛❧♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ♣r♦♣r✐❡❞❛❞❡s ❝♦♠❜✐♥❛tór✐❛s ❞❛s s✉❛s s♦❧✉çõ❡s ót✐♠❛s ❡ s❡✉s ❛s♣❡❝t♦s ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❝♦♠♣✉t❛❝✐♦♥❛❧✳ ❯♠❛ r❡✈✐sã♦ ❜✐❜❧✐♦❣rá✜❝❛ ❝♦♠ ♦s tr❛❜❛❧❤♦s q✉❡ ❥á ❛❜♦r❞❛r❛♠ ♦ ♣r♦❜❧❡♠❛ ❡ s✉❛s ✈❛r✐❛çõ❡s ❝♦rr❡❧❛t❛s t❛♠❜é♠ é ❛♣r❡s❡♥t❛❞❛✳

❈❛♣ít✉❧♦ ✸✳ ❬❋♦r♠✉❧❛çõ❡s ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ♣❛r❛ ♦ P❙❇▼●❪

❋♦r♠✉❧❛çõ❡s ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ♣❛r❛ ♦ P❙❇▼● sã♦ ✐♥tr♦❞✉③✐❞❛s✳ ❙ã♦ t❛♠❜é♠ ❛♣r❡s❡♥t❛❞❛s ❞✉❛s ♥♦✈❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❧✐♥❡❛r❡s ✈á❧✐❞❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳

❈❛♣ít✉❧♦ ✹✳ ❬❯♠ ▼ét♦❞♦ ❞❡ ❙♦❧✉çã♦ ❊①❛t❛ ♣❛r❛ ♦ P❙❇▼●❪

❆❧❣♦r✐t♠♦s ❞❡ ♣❧❛♥♦s ❞❡ ❝♦rt❡ ♣❛r❛ ❛✈❛❧✐❛r ♦s ❧✐♠✐t❡s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ❞❡ ✉♠❛ ❞❛s ❢♦r♠✉❧❛çõ❡s ♣r♦♣♦st❛s sã♦ ✐♥tr♦❞✉③✐❞♦s ❡ ✉t✐❧✐③❛❞♦s ❡♠ ✉♠ ❛❧❣♦r✐t♠♦ ❇r❛♥❝❤✲❛♥❞✲ ❝✉t ♣❛r❛ ♦ P❙❇▼●✳

❈❛♣ít✉❧♦ ✺✳ ❬▼ét♦❞♦s ❞❡ ❙♦❧✉çã♦ ❍❡✉ríst✐❝❛ ♣❛r❛ ♦ P❙❇▼●❪

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✹ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦

♣r♦♣♦st❛s ❡ ❛✈❛❧✐❛❞❛s ♣❛r❛ ♦ P❙❇▼● ✐s♦❧❛❞❛♠❡♥t❡ ❡ ❡♠ ❝♦♥❥✉♥çã♦ ❝♦♠ ♦ ❛❧❣♦r✐t♠♦ ❇r❛♥❝❤✲❛♥❞✲❝✉t ❞✐s❝✉t✐❞♦ ♥♦ ❈❛♣ít✉❧♦ ✹✳

❈❛♣ít✉❧♦ ✻✳ ❬❈♦♥❝❧✉sõ❡s ❡ ❚r❛❜❛❧❤♦s ❋✉t✉r♦s❪

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❈❛♣ít✉❧♦ ✷

❖ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦

▼í♥✐♠♦ ●❡♥❡r❛❧✐③❛❞♦

◆❡st❡ ❝❛♣ít✉❧♦✱ ✐♥tr♦❞✉③✐♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ❛ ♥♦t❛çã♦ ♥❡❝❡ssár✐❛s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳ ❉❡✜♥✐♠♦s ❢♦r♠❛❧♠❡♥t❡ ♦ P❙❇▼● ❡ ♦✉tr♦s ♣r♦❜❧❡✲ ♠❛s ❛ ❡❧❡ r❡❧❛❝✐♦♥❛❞♦s✳ ❋❛③❡♠♦s t❛♠❜é♠ ✉♠❛ r❡✈✐sã♦ ❜✐❜❧✐♦❣rá✜❝❛ ❞❡ tr❛❜❛❧❤♦s q✉❡ ❥á ❛❜♦r❞❛r❛♠ ♦ P❙❇▼●✱ ♦✉ ❛❧❣✉♠❛s ✈❛r✐❛çõ❡s ❞♦ ♠❡s♠♦✱ ✈✐❛ ❛❧❣♦r✐t♠♦s ❡①❛t♦s✱ ❛✲ ♣r♦①✐♠❛t✐✈♦s ❡ ❤❡✉ríst✐❝♦s✳ ❊♠ s❡❣✉✐❞❛✱ ❞✐s❝✉t✐♠♦s s❡✉s ❛s♣❡❝t♦s ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡ s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❝♦♠❜✐♥❛tór✐❛s✳

✷✳✶ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s

P❛r❛ ❝♦♥❝❡✐t♦s ❡❧❡♠❡♥t❛r❡s ❞❡ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s✱ ✉s❛♠♦s ❝♦♠♦ r❡❢❡rê♥❝✐❛s ♦s ❧✐✈r♦s ❬❲❡st✱ ✷✵✵✶❀ ❇♦❧❧♦❜ás✱ ✶✾✾✽❀ ❉✐❡st❡❧✱ ✷✵✵✺❪✳ ❆♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❛ ♠❡♥♦s q✉❛♥❞♦ ❞✐t♦ ❡①♣❧✐❝✐t❛♠❡♥t❡✱ t♦❞❛ ♠❡♥çã♦ ❛♦ t❡r♠♦ ❣r❛❢♦ s✐❣♥✐✜❝❛ ✉♠ ❣r❛❢♦ ✜♥✐t♦✱ s✐♠♣❧❡s ❡ ♥ã♦✲❞✐r❡❝✐♦♥❛❞♦✳ ❉❡♥♦t❛♠♦s ✉♠ ❣r❛❢♦ ♣♦r ✉♠ ♣❛r ♦r❞❡♥❛❞♦ G= (V, E)✱ ❡♠ q✉❡ V é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ❡ E é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s✳ ◗✉❛♥❞♦ V ❡ E ♥ã♦ ❡st✐✈❡r❡♠ s✉❜t❡♥❞✐❞♦s✱ ❡s❝r❡✈❡♠♦s V(G) ❡ E(G) ♣❛r❛ ❛❧✉❞✐r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛♦ ❝♦♥❥✉♥t♦ ❞❡

✈ért✐❝❡s ❡ ❛r❡st❛s ❞❡ G✳ ❯s❛♠♦s ❛ ♣❛❧❛✈r❛ ❞✐❣r❛❢♦ ♣❛r❛ ♥♦s r❡❢❡r✐r ❛ ✉♠ ❣r❛❢♦ ❞✐r❡✲ ❝✐♦♥❛❞♦✳ P❛r❛ ✜♥s ❞❡ ❝❧❛r❡③❛✱ r❡s❡r✈❛♠♦s ♦ t❡r♠♦ ❛r❝♦ ❝♦♠♦ ✉♠ s✐♥ô♥✐♠♦ ❞❡ ❛r❡st❛ ❞✐r❡❝✐♦♥❛❞❛✳

❉✐③❡♠♦s q✉❡Gé ✈ért✐❝❡✲❜✐❝♦♥❡①♦ s❡ ❡①✐st❡♠ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ❝❛♠✐♥❤♦s ✈ért✐❝❡✲ ❞✐s❥✉♥t♦s ❡♥tr❡ q✉❛❧q✉❡r ♣❛r ❞❡ ✈ért✐❝❡s ❞✐st✐♥t♦s ❞❡ G✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❣r❛❢♦ ❛r❡st❛✲ ❜✐❝♦♥❡①♦ é ❞❡✜♥✐❞♦ ❛♥❛❧♦❣❛♠❡♥t❡✳ ❉✐③❡♠♦s q✉❡ u_∈V é ✉♠ ♣♦♥t♦ ❞❡ ❛rt✐❝✉❧❛çã♦ ❞❡ G s❡ ❛ s✉❛ r❡♠♦çã♦ r❡s✉❧t❛ ❡♠ ✉♠ ❣r❛❢♦ ❞❡s❝♦♥❡①♦✳ ❙❡ ♥ã♦ ❤♦✉✈❡r ❞ú✈✐❞❛ ♥♦ ❝♦♥t❡①t♦✱ ❛ ♣❛❧❛✈r❛ ❜✐❝♦♥❡①♦ s❡rá ✉s❛❞❛ ❞❛q✉✐ ❡♠ ❞✐❛♥t❡ ♣❛r❛ ❞❡s✐❣♥❛r q✉❛❧q✉❡r ❣r❛❢♦

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✻ ❈❛♣ít✉❧♦ ✷✳ ❖ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦ ▼í♥✐♠♦ ●❡♥❡r❛❧✐③❛❞♦

✈ért✐❝❡✲❜✐❝♦♥❡①♦✳

P❛r❛ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ X _⊂ V✱ ❝♦♠ X ₆= _∅✱ ♦ ❝♦rt❡ ✐♥❞✉③✐❞♦ ♣♦r X ❡♠ G✱ ❞❡♥♦t❛❞♦ ♣♦r δ(X)✱ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ❞❡ G ❝♦♠ ❡①❛t❛♠❡♥t❡ ✉♠❛ ❡①tr❡♠✐❞❛❞❡ ❡♠ X✱ ✐✳❡✳✱ δ(X) = _{{u, v_{} ∈}E :u_∈X, v _6∈X_}✳ ◗✉❛♥❞♦ ♥❡❝❡ssár✐♦✱ ♣❛r❛ ❡♥❢❛t✐③❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡♠ G✱ ❡s❝r❡✈❡♠♦s δG(X) ❡♠ ✈❡③ q✉❡ δ(X)✳ ❯♠❛ d✲♣❛rt✐çã♦ ❞❡ ✉♠

❝♦♥❥✉♥t♦ ✜♥✐t♦W ₆=∅✱ ♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦ d > 0✱ é ✉♠ ❝♦♥❥✉♥t♦ P ={P1, . . . , Pd} ❞❡

❛❣❧♦♠❡r❛❞♦s ✭♦✉ ❝❧✉st❡rs✮ ❡♠ q✉❡Pi ⊂W✱Pi 6=∅♣❛r❛ t♦❞♦i∈ {1, . . . , d}✱Pi∩Pj =∅

♣❛r❛ q✉❛✐sq✉❡r i, j _{∈ {}1, . . . , d_} ❞✐st✐♥t♦s ❡_∪d

i=1Pi =W✳

P❛r❛ ❝♦♥❝❡✐t♦s ❡❧❡♠❡♥t❛r❡s ❞❡ ❚❡♦r✐❛ ❞❛ ❈♦♠♣❧❡①✐❞❛❞❡ ❈♦♠♣✉t❛❝✐♦♥❛❧✱ ♥ós ♥♦s ❛♣♦✐❛♠♦s ♥❛s r❡❢❡rê♥❝✐❛s ❬❏♦❤♥s♦♥ ✫ ●❛r❡②✱ ✶✾✼✾❀ ❆r♦r❛ ✫ ❇❛r❛❦✱ ✷✵✵✼❪✳ P❛r❛ ✉♠ ♥ú♠❡r♦ r❡❛❧ σ _≥ 1 ❡ ✉♠ ❞❛❞♦ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛çã♦ ❞❡ ♠✐♥✐♠✐③❛çã♦ ❝♦♠ s♦❧✉çã♦

ót✐♠❛ ❞❡ ❝✉st♦w✱ ✉♠ ❛❧❣♦r✐t♠♦ σ✲❛♣r♦①✐♠❛t✐✈♦ ♣❛r❛ ❡ss❡ ♣r♦❜❧❡♠❛✱ ♦✉ ✉♠ ❛❧❣♦✲ r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♠ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦ ♦✉ ❣❛r❛♥t✐❛ ❞❡ ❞❡s❡♠♣❡♥❤♦ σ✱ é ✉♠ ❛❧❣♦r✐t♠♦ q✉❡ ❡♥❝♦♥tr❛✱ ♣❛r❛ t♦❞❛ ✐♥stâ♥❝✐❛ ❞♦ ♣r♦❜❧❡♠❛✱ ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ ❝♦♠ ❝✉st♦✱ ♥♦ ♠á①✐♠♦✱ ✐❣✉❛❧ ❛σw ❡✱ ♥♦ ♣✐♦r ❝❛s♦✱ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ ♥♦ t❛♠❛♥❤♦ ❞❛ ✐♥stâ♥❝✐❛✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❛❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦ ❞❡ ♠❛①✐♠✐③❛çã♦ é ❞❡✜♥✐❞♦ ❛♥❛❧♦❣❛♠❡♥t❡✳

❱❛♠♦s ❞❡♥♦t❛r ♣♦r R+✱ R∗+ ❡ Z+✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ❝♦♥❥✉♥t♦s ❞♦s ♥ú♠❡r♦s

r❡❛✐s ♥ã♦✲♥❡❣❛t✐✈♦s✱ ❞♦s r❡❛✐s ♣♦s✐t✐✈♦s ❡ ❞♦s ✐♥t❡✐r♦s ♥ã♦✲♥❡❣❛t✐✈♦s✳ ❯s❛♠♦s ♦ sí♠❜♦❧♦ B♣❛r❛ ❞❡♥♦t❛r ♦ ❝♦♥❥✉♥t♦ {0,1_}✳

✷✳✷ ❉❡✜♥✐çã♦ ❞♦ Pr♦❜❧❡♠❛ ❡ Pr♦❜❧❡♠❛s

❘❡❧❛❝✐♦♥❛❞♦s

▼✉✐t♦s ♣r♦❜❧❡♠❛s ❡♠ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛ ♣♦ss✉❡♠ ✉♠❛ ❡str❡✐t❛ r❡❧❛çã♦ ❝♦♠ ♦ P❙❇▼✭●✮✳ ❱✐st♦ q✉❡ ❣r❛♥❞❡ ♣❛rt❡ ❞♦ q✉❡ ❥á s❡ ❢❡③ ❡ ❥á s❡ s❛❜❡ ❛ r❡s♣❡✐t♦ ❞♦ P❙❇▼✭●✮ é ♦r✐✉♥❞♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❛ss♦❝✐❛❞♦ ❛ ❡ss❡s ♣r♦❜❧❡♠❛s✱ ♣❛r❛ ✜♥s ❞❡ ❝❧❛r❡③❛ ❡ ❝♦♠♣❧❡t✉❞❡✱ ❞❡✜♥✐♠♦s ❛ s❡❣✉✐r ❛❧❣✉♥s ♣r♦❜❧❡♠❛s r❡❧❛❝✐♦♥❛❞♦s ❛♦ P❙❇▼● q✉❡ s❡rã♦ r❡❢❡r❡♥❝✐❛❞♦s r❡♣❡t✐❞❛s ✈❡③❡s ❛♦ ❧♦♥❣♦ ❞❛ ❞✐ss❡rt❛çã♦✳

❖ ♣r✐♠❡✐r♦ ❞❡❧❡s é ❛ ❝❤❛♠❛❞❛ ✈❡rsã♦ ♥ã♦✲❣❡♥❡r❛❧✐③❛❞❛ ❞♦ P❙❇▼●✱ ❛ s❛❜❡r ♦ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦ ▼í♥✐♠♦ ✭P❙❇▼✮✳ ❉❛❞♦ ✉♠ ❣r❛❢♦ G = (V, E) ❝♦♠

♣❡s♦s r❡❛✐s ♣♦s✐t✐✈♦s {wij ∈ R∗+ : {i, j} ∈ E} ♥❛s ❛r❡st❛s✱ ♦ P❙❇▼ t❡♠ ♣♦r ♦❜❥❡t✐✈♦

❡♥❝♦♥tr❛r ✉♠ s✉❜❣r❛❢♦ ❣❡r❛❞♦r ❜✐❝♦♥❡①♦ G′ ❞❡ G✱ t❛❧ q✉❡ c(G′) s❡❥❛ ♠í♥✐♠♦✱ ♦♥❞❡

c(G′) = P_{_i,j_}∈_E₍_G′₎wij ❞❡♥♦t❛ ♦ ❝✉st♦ ❞❡ G′✳ ◗✉❛♥❞♦ ❝♦♥✈❡♥✐❡♥t❡✱ ✉s❛♠♦s t❛♠❜é♠

❛ ♥♦t❛çã♦wijG ♣❛r❛ ♥♦s r❡❢❡r✐r ❛♦ ♣❡s♦ ❞❛ ❛r❡st❛ {i, j} ❞❡G✳

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✷✳✸✳ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá❢✐❝❛ ✼

❛❜r❡✈✐❛❞❛ ♣♦r h✲P❙❇▼✳ ❉❛❞❛ ✉♠❛ ❢✉♥çã♦ h : V _→ Z+✱ ♦ h✲P❙❇▼ ♣r♦❝✉r❛ ✉♠ s✉❜✲

❣r❛❢♦ G′ ❞❡G ❝♦♠ ❝✉st♦ c(G′)♠í♥✐♠♦✱ t❛❧ q✉❡ ❡①✐st❛✭♠✮ ♣❡❧♦ ♠❡♥♦s min_{h(u), h(v)_}

❝❛♠✐♥❤♦✭s✮ ✈ért✐❝❡✲❞✐s❥✉♥t♦✭s✮ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ✈ért✐❝❡s ❞✐st✐♥t♦su, v _∈V✳ ❖❜s❡r✈❛♠♦s q✉❡ ♦ P❙❇▼ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ h✲P❙❇▼ q✉❛♥❞♦ h(u) = 2 ♣❛r❛ t♦❞♦ u_∈V✳

❖ ♣ró①✐♠♦ ♣r♦❜❧❡♠❛ q✉❡ ❛♣r❡s❡♥t❛♠♦s é ♦ Pr♦❜❧❡♠❛ ❞♦ Pr♦❥❡t♦ ❞❡ ❘❡❞❡s ❘❡✲ s✐❧✐❡♥t❡s ✭PP❘❘✮✳ ❖ PP❘❘ ♣♦❞❡ s❡r ❜r❡✈❡♠❡♥t❡ ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠❛ r❡❧❛①❛çã♦ ❝♦♠❜✐✲ ♥❛tór✐❛ ❞♦h✲P❙❇▼ q✉❡ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ❞❡ ❝✉st♦ ♠í♥✐♠♦ ❝♦♥t❡♥❞♦ ❝❛♠✐♥❤♦s ❛r❡st❛✲❞✐s❥✉♥t♦s✳ ➱ ✐♠❡❞✐❛t♦ ✈❡r q✉❡ ♦ PP❘❘ é ✉♠❛ r❡❧❛①❛çã♦ ❝♦♠❜✐♥❛tór✐❛ ❞♦ h✲P❙❇▼✱ ✉♠❛ ✈❡③ q✉❡ t♦❞❛ s♦❧✉çã♦ ✈✐á✈❡❧ ♣❛r❛ ♦h✲P❙❇▼ é t❛♠❜é♠ ✈✐á✈❡❧ ♣❛r❛ ♦ PP❘❘✳ ◗✉❛♥❞♦ h(u) = 2 ♣❛r❛ t♦❞♦ ✈ért✐❝❡ u _∈ V✱ ♦ PP❘❘ é t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ♥❛ ❧✐t❡r❛t✉r❛ ♣❡❧♦ ♥♦♠❡ ❞❡ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❆r❡st❛✲❇✐❝♦♥❡①♦ ▼í♥✐♠♦ ✭P❙❆❇▼✮✳

P♦❞❡♠♦s ❛❣♦r❛ ❞❡✜♥✐r ❢♦r♠❛❧♠❡♥t❡ ♦ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦ ▼í♥✐♠♦ ●❡♥❡r❛❧✐③❛❞♦ ✭P❙❇▼●✮✳ ❙❡❥❛ Q =_{V1, . . . , Vk}✱ ♣❛r❛ k =|Q|> 2✱ ✉♠❛ k✲♣❛rt✐çã♦ ❞❡

V✳ ❖ P❙❇▼● ❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r ✉♠ s✉❜❣r❛❢♦ ❜✐❝♦♥❡①♦G′ ❞❡G ❝♦♠ ❝✉st♦c(G′)

♠í♥✐♠♦ ❡ ❝♦♥t❡♥❞♦ ❡①❛t❛♠❡♥t❡ ✉♠ ✈ért✐❝❡ ❞❡ ❝❛❞❛ ❝❧✉st❡r ❞❡ Q✳ ❉❛❞❛ ✉♠❛ ✐♥stâ♥❝✐❛ q✉❛❧q✉❡r ❞♦ P❙❇▼● ❡ ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ S = (VS, ES) ♣❛r❛ ❡ss❛ ✐♥stâ♥❝✐❛✱ ❞✐③❡♠♦s

q✉❡ u _∈ V é ✉♠ ✈ért✐❝❡ ❡❧❡✐t♦ ✭♦✉ ❝❧✉st❡r✲❤❡❛❞✮ ❡♠ S s❡ u _∈ VS✳ ❆❧é♠ ❞✐ss♦✱

✉♠❛ ❛r❡st❛ e _∈ ES é ❞✐t❛ s❡r r❡❞✉♥❞❛♥t❡ ❡♠ r❡❧❛çã♦ ❛ S s❡ s✉❛ r❡♠♦çã♦ ♣r❡s❡r✈❛

❛ ✈✐❛❜✐❧✐❞❛❞❡ ❡ ❞✐♠✐♥✉✐ ♦ ❝✉st♦ ❞❡ S✳ ❉❛q✉✐ ❡♠ ❞✐❛♥t❡✱ ✈❛♠♦s ♥♦s r❡❢❡r✐r ❛ q✉❛❧q✉❡r ✐♥stâ♥❝✐❛ ♦✉ s♦❧✉çã♦ ✈✐á✈❡❧ ♣❛r❛ ♦ P❙❇▼● ❡ s✉❛s ✈❛r✐❛çõ❡s ❝♦♠ ♦s sí♠❜♦❧♦s ❞❡✜♥✐❞♦s ♥❡st❡ ♣❛rá❣r❛❢♦✳

✷✳✸ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛

❊♠ ✈✐st❛ ❞❛ s✐♠✐❧❛r✐❞❛❞❡ ❡♥tr❡ ♦s ♣r♦❜❧❡♠❛s ❞❡✜♥✐❞♦s ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ ❛ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ✉♠❛ r❡✈✐sã♦ ❜✐❜❧✐♦❣rá✜❝❛ ♣❛r❛ ♦ P❙❇▼✭●✮ ✐♥❝❧✉✐♥❞♦✱ q✉❛♥❞♦ ❝♦♥✈✐r✱ tr❛❜❛❧❤♦s q✉❡ t❛♠❜é♠ ❧✐❞❛♠ ❝♦♠ ♦ PP❘❘✱ h✲P❙❇▼ ♦✉ P❙❆❇▼✳ ❆ ✜♠ ❞❡ s✐♠♣❧✐✜❝❛r ❛ ♥♦t❛çã♦✱ ✉s❛♠♦s n =_|V_| ❡ m =_|E_| ♣❛r❛ r❡♣r❡s❡♥t❛r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛s ❝❛r❞✐♥❛❧✐✲ ❞❛❞❡s ❞♦s ❝♦♥❥✉♥t♦s ❞❡ ✈ért✐❝❡s ❡ ❛r❡st❛s ❞❛ ✐♥stâ♥❝✐❛ G✳

✷✳✸✳✶ ❆❧❣♦r✐t♠♦s ❞❡ ❆♣r♦①✐♠❛çã♦

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✽ ❈❛♣ít✉❧♦ ✷✳ ❖ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❇✐❝♦♥❡①♦ ▼í♥✐♠♦ ●❡♥❡r❛❧✐③❛❞♦

♦❜té♠ s♦❧✉çõ❡s ✈✐á✈❡✐s ❝♦♠♣✉t❛♥❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ✉♠❛ ár✈♦r❡ ❞❡ ❜✉s❝❛ ❞❡ ❉❋❙ ✭❞❡♣t❤✲ ✜rst s❡❛r❝❤✮ T ❞❡G✳ ❊♠ s❡❣✉✐❞❛✱ ♦ ❛❧❣♦r✐t♠♦ ❛❣r❡❣❛ ❛T ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❛s ❛r❡st❛s ❞❡ r❡❝✉♦ ✭❜❛❝❦ ❡❞❣❡s✮✱ ❡♥❝♦♥tr❛❞❛s ❞✉r❛♥t❡ ❛ ❉❋❙✱ ♣❡q✉❡♥♦ ♦ ❜❛st❛♥t❡ ♣❛r❛ ✏❝♦❜r✐r✑ ♦s ♣♦♥t♦s ❞❡ ❛rt✐❝✉❧❛çã♦ ❡♠T✳ P♦r ✜♠✱ ♦ ❛❧❣♦r✐t♠♦ ❞❡t❡❝t❛ ❡ ♣♦❞❛ ❛r❡st❛s r❡❞✉♥❞❛♥t❡s ❞❛ s♦❧✉çã♦ ❝♦rr❡♥t❡✳ P❛r❛ ❢❛③❡r ❛ ❛♥á❧✐s❡ ❞❛ ❣❛r❛♥t✐❛ ❞❡ ❞❡s❡♠♣❡♥❤♦ ❞♦ s❡✉ ❛❧❣♦r✐t♠♦✱ ❑❤✉❧❧❡r ❡ ❱✐s❤❦✐♥ ♣r♦♣✉s❡r❛♠ ❡ ✉t✐❧✐③❛r❛♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡s❝✉❧t✉r❛ ❞❡ ✉♠ ❣r❛❢♦ ✭❣r❛♣❤ ❝❛r✈✐♥❣✮✳ ❖s ❛✉t♦r❡s r❡❛❧✐③❛r❛♠ ❡①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❡♠ ✐♥stâ♥❝✐❛s ❛❧❡❛tór✐❛s ❡ ✈❡r✐✜❝❛r❛♠ q✉❡✱ ♥❛ ♠é❞✐❛✱ ♦ ❛❧❣♦r✐t♠♦ ♣♦r ❡❧❡s ♣r♦♣♦st♦ ♦❜t❡✈❡ s♦❧✉çõ❡s ❞❡ 10% ❛ 12% ♣✐♦r❡s q✉❡ ♦ ót✐♠♦ ♣❛r❛ ❣r❛❢♦s ❡s♣❛rs♦s ❡2% ♣✐♦r❡s ♣❛r❛ ❣r❛❢♦s ❞❡♥s♦s✳

●❛r❣ ❡t ❛❧✳ ❬✶✾✾✸❪ ♠❡❧❤♦r❛r❛♠ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❑❤✉❧❧❡r✲❱✐s❤❦✐♥ ❛♣r❡s❡♥t❛♥❞♦ ✉♠ ♥♦✈♦ ❛❧❣♦r✐t♠♦ O(n+m) 3

2✲❛♣r♦①✐♠❛t✐✈♦✱ q✉❡ ♦♣❡r❛ ❡♠ ❞♦✐s ❡stá❣✐♦s✳ ❙❡✉ ♣r✐♠❡✐r♦

❡stá❣✐♦ é ✐❞ê♥t✐❝♦ ❛♦ ❛❧❣♦r✐t♠♦ ❞❡ ❑❤✉❧❧❡r✲❱✐s❤❦✐♥✳ ◆♦ s❡❣✉♥❞♦ ❡stá❣✐♦✱ ❡♠♣r❡❣❛✲s❡ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ♣❛rt✐❝✐♦♥❛r ♦s ✈ért✐❝❡s ❞❡ G ❡♠ ❜❧♦❝♦s ❡✱ ❡♠ s❡❣✉✐❞❛✱ sã♦ ✐❞❡♥t✐✜✲ ❝❛❞❛s ❡ r❡t✐r❛❞❛s ❛r❡st❛s r❡❞✉♥❞❛♥t❡s ❞❡ ❝❛❞❛ ❜❧♦❝♦✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❛s ❝❤❛♠❛❞❛s ❛r❡st❛s ❝♦r❞❛s✱ ✐✳❡✳✱ ❛r❡st❛s q✉❡ ❧✐❣❛♠ ✈ért✐❝❡s ♥ã♦✲❛❞❥❛❝❡♥t❡s ❞❡ ❝✐❝❧♦s ❝♦♠ ♠❛✐s ❞❡ três ✈ért✐❝❡s ❡♠ G✳ ❆ ❛♥á❧✐s❡ ❞♦ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❢♦✐ ❢❡✐t❛ ✉s❛♥❞♦ ❝♦♠♦ s✉♣♦rt❡ ❛ ♦❜s❡r✈❛çã♦ ❞❡ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡G❝♦♠i✈ért✐❝❡s✱ t♦❞♦ s✉❜❣r❛❢♦ ❣❡r❛❞♦r ❜✐❝♦♥❡①♦ ❞❡G t❡♠✱ ♥♦ ♠í♥✐♠♦✱max_{n,2i_} ❛r❡st❛s✳ ❖s ❛✉t♦r❡s✱ ♣♦r ✜♠✱ ♣r♦✈❛r❛♠ q✉❡ ♦ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦ ❞❡ 3

2 é ❥✉st♦ ❡①♣❧✐❝✐t❛♥❞♦ ❝❧❛ss❡s ❞❡

✐♥stâ♥❝✐❛s ❞♦ P❙❇▼ ♣❛r❛ ❛s q✉❛✐s ❡ss❡ ❢❛t♦r é ❛t✐♥❣✐❞♦✳

❊♠ ❬❱❡♠♣❛❧❛ ✫ ❱❡tt❛✱ ✷✵✵✵❪✱ ❢♦✐ ♣r♦♣♦st♦ ✉♠ ❛❧❣♦r✐t♠♦ 4

3✲❛♣r♦①✐♠❛t✐✈♦ ❝♦♠

❝♦♠♣❧❡①✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ❞❡ t❡♠♣♦ ♥♦ ♣✐♦r ❝❛s♦ ❞❡O(n2m)✱ q✉❡ é ♦ ♠❡❧❤♦r ❝♦♥❤❡❝✐❞♦

♣❛r❛ ♦ P❙❇▼ ❝♦♠ ♣❡s♦s ✉♥✐❢♦r♠❡s ❡♠ t❡r♠♦s ❞❡ ❣❛r❛♥t✐❛ ❞❡ ❞❡s❡♠♣❡♥❤♦✳ ❖s ❛✉t♦r❡s ❝♦♥st❛t❛r❛♠ q✉❡ ♦ ♠❡♥♦r s✉❜❣r❛❢♦ ❣❡r❛❞♦r ❛r❡st❛✲❜✐❝♦♥❡①♦ ❞❡ G ❝♦♠ ❣r❛✉ ♠í♥✐♠♦ ✐❣✉❛❧ ❛2♣r♦✈ê ✉♠ ❧✐♠✐t❡ ✐♥❢❡r✐♦r ♣❛r❛ ❛ s♦❧✉çã♦ ót✐♠❛ ❞♦ P❙❇▼ ❝♦♠ ♣❡s♦s ✉♥✐❢♦r♠❡s✳

❱❡♠♣❛❧❛ ❡ ❱❡tt❛ ❝❤❛♠❛r❛♠ ❞❡ ❉✷ ♦ ♣r♦❜❧❡♠❛ ❞❡ s❡ ♦❜t❡r ❡ss❡ ❧✐♠✐t❡ ✐♥❢❡r✐♦r ❡ ❞❡♠♦♥s✲ tr❛r❛♠ q✉❡ ❡❧❡ s❡ r❡❞✉③ ❛♦ Pr♦❜❧❡♠❛ ❞♦ ❊♠♣❛r❡❧❤❛♠❡♥t♦ ❞❡ ❈❛r❞✐♥❛❧✐❞❛❞❡ ▼á①✐♠❛✳ ❊♠ ❧✐♥❤❛s ❣❡r❛✐s✱ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❱❡♠♣❛❧❛✲❱❡tt❛ ♣r✐♠❡✐r❛♠❡♥t❡ r❡s♦❧✈❡ ❉✷ ✉s❛♥❞♦ ✉♠❛ ♠♦❞✐✜❝❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦O(m√n)❞❡ ▼✐❝❛❧✐✲❱❛③✐r❛♥✐ ❬▼✐❝❛❧✐ ✫ ❱❛③✐r❛♥✐✱ ✶✾✽✵❪✱ ❛

✐♠♣❧❡♠❡♥t❛çã♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ♠❛✐s rá♣✐❞❛ q✉❡ s❡ ❝♦♥❤❡❝❡ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊❞♠♦♥❞s ❬❊❞♠♦♥❞s✱ ✶✾✻✺❜❪✳ ❊♠ s❡❣✉✐❞❛✱ ❡❧❡ ❝✉✐❞❛❞♦s❛♠❡♥t❡ ❛❞✐❝✐♦♥❛ ❛r❡st❛s à s♦❧✉çã♦ ❞❡ ❉✷ ♣❛r❛ ❛ss❡❣✉r❛r q✉❡ ♥ã♦ ❡①✐st❛♠ ♠❛✐s ♣♦♥t♦s ❞❡ ❛rt✐❝✉❧❛çã♦✳ ❖ ♠ér✐t♦ ❞♦ tr❛❜❛❧❤♦ ❡♠ q✉❡stã♦ ❢♦✐ ❥✉st❛♠❡♥t❡ ♣r♦✈❛r q✉❡ é ♣♦ssí✈❡❧ ✏✈ért✐❝❡✲❜✐❝♦♥❡❝t❛r✑ ❛ s♦❧✉çã♦ ❞❡ ❉✷✱ ✐✳❡✳✱ tr❛♥s❢♦r♠❛r ❛ s♦❧✉çã♦ ❞❡ ❉✷ ❡♠ ✉♠ ❣r❛❢♦ ✈ért✐❝❡✲❜✐❝♦♥❡①♦✱ ✉s❛♥❞♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❛s ❛r❡st❛s ❞❡G ❝♦♠ ❝❛r❞✐♥❛❧✐❞❛❞❡ ♥ã♦ ♠❛✐♦r q✉❡ ✉♠ t❡rç♦ ❞❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❛s ❛r❡st❛s ❞❡ ❉✷✳

(31)

✷✳✸✳ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá❢✐❝❛ ✾

❬✶✾✽✶❪ ❞❡s❡♥✈♦❧✈❡r❛♠ ✉♠ ❛❧❣♦r✐t♠♦ O(n2) 3✲❛♣r♦①✐♠❛t✐✈♦✳ ❆ ❞❡s❝r✐çã♦ ❞♦ ❛❧❣♦r✐t♠♦

❞❡ ❋r❡❞❡r✐❝❦s♦♥✲❏❛❏❛ é ❜❡♠ ❝♦♠♣❧❡①❛ ❡ ❡①t❡♥s❛✱ ♣♦ré♠✱ ❡♠ ❛❧t♦ ♥í✈❡❧✱ t❛❧ ❛❧❣♦r✐t♠♦ ✐♥✐❝✐❛❧♠❡♥t❡ ♦❜té♠ ✉♠❛ ár✈♦r❡ ❣❡r❛❞♦r❛ ♠í♥✐♠❛ ❞❡G❡✱ ❡♠ s❡❣✉✐❞❛✱ ✈ért✐❝❡✲❜✐❝♦♥❡❝t❛ ❡ss❛ ár✈♦r❡ ❛♥❛❧✐s❛♥❞♦ ❛ ❡str✉t✉r❛ t♦♣♦❧ó❣✐❝❛ ❞❛ s✉❛ ❝❤❛♠❛❞❛ ár✈♦r❡ ❞❡ ❜❧♦❝♦s ❡ ♣♦♥t♦s ❞❡ ❛rt✐❝✉❧❛çã♦ ✭❜❧♦❝❦✲❝✉t✈❡rt❡① tr❡❡✮✳

❆ ♠❡s♠❛ ❣❛r❛♥t✐❛ ❞❡ ❞❡s❡♠♣❡♥❤♦ ❢♦✐ ❛❧❝❛♥ç❛❞❛ ♣♦r ✉♠ ❛❧❣♦r✐t♠♦ O(min{m,2n_}n2₎ _{❞❡ ❘❛✈✐ ✫ ❲✐❧❧✐❛♠s♦♥ ❬✶✾✾✺❪✳ ❖ ❛❧❣♦r✐t♠♦ ❞❡ ❘❛✈✐✲❲✐❧❧✐❛♠s♦♥✱}

❝♦♠♦ q✉❛s❡ t♦❞♦s ❛té ❛q✉✐ ❞✐s❝✉t✐❞♦s✱ t❛♠❜é♠ é ❜✐✲❢ás✐❝♦✳ ◆❛ s✉❛ ♣r✐♠❡✐r❛ ❢❛s❡✱ ❡❧❡ ❡♠♣r❡❣❛ ✉♠ ❛❧❣♦r✐t♠♦ ♣r✐♠❛❧✲❞✉❛❧ ❞❡ ●♦❡♠❛♥s ✫ ❲✐❧❧✐❛♠s♦♥ ❬✶✾✾✺❛❪ ♣❛r❛ r❡s♦❧✈❡r ❡♠ O(n2logn) ❝♦♠ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦ ✐❣✉❛❧ ❛2 ♦ P❙❆❇▼✳ ◆❛ s✉❛ s❡❣✉♥❞❛ ❢❛s❡✱ ♦

❛❧❣♦r✐t♠♦ ❛✐♥❞❛ ✉t✐❧✐③❛ ✉♠ ❡sq✉❡♠❛ ♣r✐♠❛❧✲❞✉❛❧ ♣❛r❛ ❡s❝♦❧❤❡r ❡♠ G ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ❞❡ ❜❛✐①♦ ❝✉st♦ t♦t❛❧ ❛ ✜♠ ❞❡ tr❛♥s❢♦r♠❛r ❛ s♦❧✉çã♦ ❡♥❝♦♥tr❛❞❛ ♥❛ ♣r✐♠❡✐r❛ ❢❛s❡ ❡♠ ✉♠ ❣r❛❢♦ ✈ért✐❝❡✲❜✐❝♦♥❡①♦✳

◆♦ ♠❡s♠♦ ❛♥♦✱ ❑❤✉❧❧❡r ✫ ❘❛❣❤❛✈❛❝❤❛r✐ ❬✶✾✾✺❪ ❛♣r❡s❡♥t❛r❛♠ ✉♠ ❛❧❣♦r✐t♠♦ O(n2_m₎ _{❝♦♠ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦} _{2 +} 1

n

✳ ❚❛❧ ❛❧❣♦r✐t♠♦ ❝r✐❛ ✉♠❛ ❝ó♣✐❛ ❞✐r❡❝✐♦♥❛❞❛ D ❞❡ G✱ ❝♦♠ ✉♠ ✈ért✐❝❡ r❛✐③ ❛rt✐✜❝✐❛❧ r ❡ ❛r❝♦s ❛❞✐❝✐♦♥❛✐s ❝♦♠ ❝✉st♦ ♥✉❧♦ s❛✐♥❞♦ ❞❡ r ❡ ✐♥❝✐❞✐♥❞♦ s♦❜r❡ ❝❛❞❛ ✈ért✐❝❡ ❞❡ D✳ ❖ ❛❧❣♦r✐t♠♦ ❞❡ ❑❤✉❧❧❡r✲❘❛❣❤❛✈❛❝❤❛r✐ ♣r♦❝❡❞❡ ✉s❛♥❞♦ ✉♠ ❛❧❣♦r✐t♠♦ O(n2m) ♣r♦♣♦st♦ ♣♦r ❋r❛♥❦ ✫ ❚❛r❞♦s ❬✶✾✽✾❪ ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠

s✉❜❞✐❣r❛❢♦ ❣❡r❛❞♦r ❞❡D❝♦♠ ❝✉st♦ ♠í♥✐♠♦✱ t❛❧ q✉❡ ❡①✐st❛♠ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ❝❛♠✐♥❤♦s ❞✐r❡❝✐♦♥❛❞♦s ✈ért✐❝❡✲❞✐s❥✉♥t♦s ❡♥tr❡ r❡ ♦s ✈ért✐❝❡s ❡①tr❡♠♦s ❞❛ ❛r❡st❛e❞❡ ♠❡♥♦r ❝✉st♦ ❡♠ G✳ ❑❤✉❧❧❡r ✫ ❘❛❣❤❛✈❛❝❤❛r✐ ❬✶✾✾✺❪ ♣r♦✈❛r❛♠ q✉❡ ❛ s♦❧✉çã♦ ♣❛r❛ ❡ss❡ ♣r♦❜❧❡♠❛ ✉♥✐❞❛ à ❛r❡st❛ e ✐♥❞✉③ ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ ♣❛r❛ ♦ P❇❙▼ ❝✉❥♦ ❝✉st♦ é✱ ♥♦ ♠á①✐♠♦✱

2 + _n1 ✈❡③❡s ♦ ót✐♠♦✳

P♦r ✜♠✱ ❋❧❡✐s❝❤❡r ❬✷✵✵✶❪ ♣r♦❥❡t♦✉ ♦ ♠❡❧❤♦r ❛❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♥❤❡❝✐❞♦ ♣❛r❛ P❙❇▼✱ ❡♠ t❡r♠♦s ❞❡ ❣❛r❛♥t✐❛ ❞❡ ❞❡s❡♠♣❡♥❤♦✱ ❝♦♠ ✉♠ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦ ❞❡

2✳ ❖ ❛❧❣♦r✐t♠♦ ❞❡ ❋❧❡✐s❝❤❡r é ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ✐t❡r❛t✐✈♦ q✉❡ ❛ ❝❛❞❛ ✐t❡r❛çã♦ r❡s♦❧✈❡

❛ r❡❧❛①❛çã♦ ❧✐♥❡❛r ❞❡ ✉♠❛ ❢♦r♠✉❧❛çã♦ ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ♣❛r❛ ♦ P❙❇▼✳ ❯♠❛ ✈❡③ ❝❛❧❝✉❧❛❞❛ ❛ s♦❧✉çã♦ ót✐♠❛ ❞♦ ♣r♦❣r❛♠❛ ❧✐♥❡❛r ❞❡✜♥✐❞♦ ❡♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ✐t❡r❛çã♦✱ t♦❞❛s ❛s ❛r❡st❛s ❞❡ G ❝✉❥❛s ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦ ❛ss♦❝✐❛❞❛s ❛ss✉♠✐r❡♠ ✈❛❧♦r ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ 1

2 sã♦ r❡t✐r❛❞❛s ❞♦ ❣r❛❢♦ ❞❡ ❡♥tr❛❞❛ ❡ ♣❛ss❛♠ ❝♦♠♣♦r ❛ s♦❧✉çã♦ ❝♦rr❡♥t❡✳ ■ss♦ s❡

r❡♣❡t❡ ❛té ♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ❛ s♦❧✉çã♦ ❝♦rr❡♥t❡ s❡ t♦r♥❛ ✈✐á✈❡❧ ♣❛r❛ ♦ P❙❇▼✳ ◆❡ss❡ ❝❛s♦✱ ❋❧❡✐s❝❤❡r ♠♦str♦✉ q✉❡ ♦ ❝✉st♦ ❞❡ss❛ s♦❧✉çã♦ ♥ã♦ ✉❧tr❛♣❛ss❛ ♦ ❞♦❜r♦ ❞♦ ót✐♠♦✳