❆▲●❖❘■❚▼❖❙ P❆❘❆ ❖ P❘❖❇▲❊▼❆ ❉❖
❙❯❇●❘❆❋❖ ❆❈❮❈▲■❈❖ ▼➪❳■▼❖ ❙❖❇
❙❮▲❱■❆ ▼❆❘■❆ ❙❆◆❚❆◆❆ ▼❆P❆
❆▲●❖❘■❚▼❖❙ P❆❘❆ ❖ P❘❖❇▲❊▼❆ ❉❖
❙❯❇●❘❆❋❖ ❆❈❮❈▲■❈❖ ▼➪❳■▼❖ ❙❖❇
❘❊❙❚❘■➬Õ❊❙ ❉■❙❏❯◆❚■❱❆❙
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ✲●r❛❞✉❛çã♦ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦ ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐✲ ❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ❝♦♠♦ r❡q✉✐✲ s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❉♦✉✲ t♦r ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦✳
❖r✐❡♥t❛❞♦r✿ ❙❡❜❛st✐á♥ ❆❧❜❡rt♦ ❯rr✉t✐❛
❝
✷✵✶✹✱ ❙í❧✈✐❛ ▼❛r✐❛ ❙❛♥t❛♥❛ ▼❛♣❛✳ ❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳
❙❛♥t❛♥❛ ▼❛♣❛✱ ❙í❧✈✐❛ ▼❛r✐❛
▼✷✾✼❛ ❆❧❣♦r✐t♠♦s ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✈❛s ✴ ❙í❧✈✐❛ ▼❛r✐❛ ❙❛♥t❛♥❛ ▼❛♣❛✳ ✖ ❇❡❧♦ ❍♦r✐③♦♥t❡ ✲ ▼✐♥❛s ●❡r❛✐s✱ ✷✵✶✹
①①✐✱ ✼✼ ❢✳ ✿ ✐❧✳ ❀ ✷✾❝♠
❚❡s❡ ✭❞♦✉t♦r❛❞♦✮ ✖ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s
❖r✐❡♥t❛❞♦r✿ ❙❡❜❛st✐á♥ ❆❧❜❡rt♦ ❯rr✉t✐❛
✶✳ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛✳ ✷✳ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦✳ ✸✳ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✈❛s✳ ✹✳ ❆❧❣♦r✐t♠♦s✳ ■✳ ❚ít✉❧♦✳
❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦ ♣r♦❢❡ss♦r ❙❡❜❛st✐á♥ ❆❧❜❡rt♦ ❯rr✉t✐❛✱ ♣♦r t♦❞♦ ♦ t❡♠♣♦ ❞❡❞✐❝❛❞♦ à ♦r✐❡♥t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ ❡♠ ♥♦ss♦s ❡♥❝♦♥tr♦s s❡♠❛♥❛✐s✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ♣❡❧❛ ❛♠✐③❛❞❡✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ✐♠❡♥s❛♠❡♥t❡ à ♦♣♦rt✉♥✐❞❛❞❡ ❝♦♥❝❡❞✐❞❛ ♣❡❧♦ ♣r♦❥❡t♦ ❞❡ ❞♦✉t♦r❛❞♦ ✐♥t❡r✐♥s✲ t✐t✉❝✐♦♥❛❧ ✭❉■◆❚❊❘✮✱ r❡❛❧✐③❛❞♦ ❡♥tr❡ ♦ ■♥st✐t✉t♦ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ✭■❋▼●✮ ❡♠ ♣❛r❝❡r✐❛ ❝♦♠ ❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ✭❯❋▼●✮✱ q✉❡ ❝✉❧♠✐♥♦✉ ♥❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❡ ❛ t♦❞❛s ❛s ♣❡ss♦❛s ❡♥✈♦❧✈✐❞❛s ♥♦ ♠❡s♠♦✳
❆♦s ♠❡✉s ♣❛✐s✱ ❙ô♥✐❛ ❡ ▲✉✐③✱ ♦❜r✐❣❛❞❛ ♣♦r ♠❡ ❛♣♦✐❛r❡♠ s❡♠♣r❡ ♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ♠❛✐s ♣r❡❝✐s❡✐✳ ❆♦s ♠❡✉s ✐r♠ã♦s✱ ▼✐❝❤❡❧❧❡ ❡ ❉❛♥✐❧♦✱ ♦❜r✐❣❛❞❛ ♣♦r ❛❝r❡❞✐t❛r❡♠ s❡♠♣r❡ ❡♠ ♠✐♠ ❡ ♠❡ ❞❛r❡♠ ❢♦rç❛s ♣❛r❛ ✈❡♥❝❡r ❡st❛ ❡ ♦✉tr❛s ❜❛t❛❧❤❛s✳ ❆♦ ♠❡✉ ♠❛r✐❞♦ ❆❧❜❛♥♦✱ ♥ã♦ t❡♥❤♦ ❝♦♠♦ ❛❣r❛❞❡❝❡r ♣♦r t♦❞❛ ❛ ♣❛❝✐ê♥❝✐❛✱ ❛♣♦✐♦✱ ❛❜❞✐❝❛çã♦ ❞❡ s❡✉s ✐❞❡❛✐s ❡♠ ♣r♦❧ ❞♦s ♠❡✉s✱ ❡♥✜♠✱ ♣♦r t✉❞♦ ♦ q✉❡ ♣❛ss❛♠♦s ❥✉♥t♦s ♥❡st❡s q✉❛tr♦ ú❧t✐♠♦s ❛♥♦s✱ ❜♦♥s ❡ ♥ã♦ tã♦ ❜♦♥s ♠♦♠❡♥t♦s✳ ▼✉✐t♦ ♦❜r✐❣❛❞❛✦
◆♦✈❛♠❡♥t❡✱ ❛♦ ♣r♦❢❡ss♦r ❙❡❜❛st✐á♥ ❆❧❜❡rt♦ ❯rr✉t✐❛✱ ♠❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s ♣♦r t♦❞❛ ❛ ❛❥✉❞❛✱ ♣❛❝✐ê♥❝✐❛ ❡ ❞❡❞✐❝❛çã♦✳ ❈♦st✉♠♦ ❞✐③❡r q✉❡ s♦✉ ✉♠❛ ♣❡ss♦❛ ❞❡ s♦rt❡✱ ♣♦r t❡r t✐❞♦ s❡♠♣r❡ ♦r✐❡♥t❛❞♦r❡s ❞❡ tã♦ ❛❧t❛ q✉❛❧✐❞❛❞❡✱ ❛ ❝♦♠❡ç❛r ♣❡❧♦ ♣r♦❢❡ss♦r ▼❛r❝♦♥❡ ❏❛♠✐❧s♦♥ ❋r❡✐t❛s ❙♦✉③❛✱ ♠❡✉ ♦r✐❡♥t❛❞♦r ❞❡ tr❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❞❡ ❣r❛❞✉❛çã♦ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❖✉r♦ Pr❡t♦ ✭❯❋❖P✮✱ q✉❡ ♠❡ ✐♥✐❝✐♦✉ ❝♦♠ t❛♥t❛ ❞❡✈♦çã♦ ♥❛ ár❡❛ ❞❛ P❡sq✉✐s❛ ❖♣❡r❛❝✐♦♥❛❧✳
●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r t❛♠❜é♠ ❛♦s ❝♦❧❡❣❛s ❡ ❛♠✐❣♦s ❞♦ ▲❛P❖✱ ♣♦r t❡r❡♠ ♠❡ ❛❝♦❧❤✐❞♦ tã♦ ❜❡♠ ❡ ♣r♦♣♦r❝✐♦♥❛❞♦ ✉♠ ❛♠❜✐❡♥t❡ ❞❡ tr❛❜❛❧❤♦ s❛✉❞á✈❡❧✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ P❤✐❧❧✐♣♣❡ ❙❛♠❡r✱ ♣♦r t❡r ♠❡ ❛❥✉❞❛❞♦ ♥❛ ❢❛s❡ ✐♥✐❝✐❛❧ ❞❡ ❛❞❛♣t❛çã♦✳ ❆♦ ❉❈❈ ❡ à s❡❝r❡t❛r✐❛✱ ♣❡❧♦ ❛♣♦✐♦ ❝♦♥❝❡❞✐❞♦✱ ❡ à ❈❆P❊❙✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳ P♦r ✜♠✱ ❛♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛✱ ♣r♦❢❡ss♦r❛s ❘♦s✐❛♥❡ ❡ ❖❧❣❛ ❡ ♣r♦❢❡ss♦r❡s ❘♦❜s♦♥ ❡ ❈❡❧s♦ ❘✐❜❡✐r♦✱ ♣❡❧❛s ♦❜s❡r✈❛çõ❡s q✉❡ ❡♥❣r❛♥❞❡❝❡r❛♠ ❡st❡ tr❛❜❛❧❤♦✳
✏❖ ♣♦❡♠❛✱ ❛♦ s❡r ❢❡✐t♦✱ ❞❡✈❡ ♠✉❞❛r ❛❧❣✉♠❛ ❝♦✐s❛✱ ♥❡♠ q✉❡ s❡❥❛ ❛♣❡♥❛s ♦ ♣ró♣r✐♦ ♣♦❡t❛✳ ❙❡✱ ❛♦ ♣♦❡t❛✱ ❞❡♣♦✐s ❞❡ ❢❛③❡r ♦ ♣♦❡♠❛✱ r❡st❛ ♦ ♠❡s♠♦ q✉❡ ❛♥t❡s✱ ♦ ♣♦❡♠❛ ♥ã♦ t❡rá s❡♥t✐❞♦✳✑ ✭❋❡rr❡✐r❛ ●✉❧❧❛r✮
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ sã♦ ❛❜♦r❞❛❞♦s ♦s ♣r♦❜❧❡♠❛s ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡s✲ tr✐çõ❡s ❉✐s❥✉♥t✐✈❛s ◆❡❣❛t✐✈❛s ✭❙❆▼❘❉◆✮ ❡ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✈❛s P♦s✐t✐✈❛s ✭❙❆▼❘❉P✮✱ ❛♠❜♦s ♣❡rt❡♥❝❡♥t❡s à ❝❧❛ss❡ N P✕❉✐❢í❝✐❧✳ ❉❛❞♦ ✉♠ ❝❡rt♦ ♣❛r ❞❡ ❡♥t✐❞❛❞❡s✱ ♣❛r❛ ❛s r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s ♥❡❣❛t✐✈❛s✱ ♥♦ ♠á①✐♠♦ ✉♠❛ ❞❡❧❛s ♣♦❞❡rá ❝♦♠♣♦r ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧✳ ❏á ♣❛r❛ ❛s r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s ♣♦s✐t✐✈❛s✱ ❞❛❞♦ ✉♠ ♣❛r ❞❡ ❡♥t✐❞❛❞❡s✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❡❧❛s ❞❡✈❡rá ❝♦♠♣♦r ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧✳
❖s ♣r♦❜❧❡♠❛s ❙❆▼❘❉◆ ❡ ❙❆▼❘❉P ♣♦ss✉❡♠ ❝♦♠♦ ✐♥stâ♥❝✐❛ ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦✲ ♥❛❞♦ G= (V, A) ❡ ✉♠ ❣r❛❢♦ ♥ã♦ ❞✐r❡❝✐♦♥❛❞♦ G¯ = (A, E)✱ q✉❡ r❡♣r❡s❡♥t❛ ❛s r❡str✐çõ❡s
❞✐s❥✉♥t✐✈❛s✱ ♣♦s✐t✐✈❛s ♦✉ ♥❡❣❛t✐✈❛s✱ s♦❜ ♣❛r❡s ❞❡ ❛r❝♦s ❞♦ ❣r❛❢♦ G✳ ❙❆▼❘❉◆ ❝♦♥s✐st❡
❡♠ ❞❡t❡r♠✐♥❛r ✉♠ s✉❜❝♦♥❥✉♥t♦ ♠á①✐♠♦ A′
⊆ A ♣❛r❛ ♦ q✉❛❧ ♦ s✉❜❣r❛❢♦ G′
= (V, A′
)
s❡❥❛ ❛❝í❝❧✐❝♦ ❡ A′ s❡❥❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ❡♠ ¯
Gt❛❧ q✉❡ ♥ã♦ ❡①✐st❛♠ ❛r❡st❛s ❡♥tr❡
❞♦✐s ❞❡ s❡✉s ✈ért✐❝❡s✱ ♦✉ s❡❥❛✱ ✉♠ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❡♠ G¯✳ ❙❆▼❘❉P ❝♦♥s✐st❡ ❡♠
❞❡t❡r♠✐♥❛r ✉♠ s✉❜❝♦♥❥✉♥t♦ ♠á①✐♠♦ A′ ⊆
A ♣❛r❛ ♦ q✉❛❧ ♦ s✉❜❣r❛❢♦ G′
= (V, A′
) s❡❥❛
❛❝í❝❧✐❝♦ ❡ A′ s❡❥❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ❡♠ ¯
Gt❛❧ q✉❡ t♦❞❛ ❛r❡st❛ ❡♠ E s❡❥❛ ✐♥❝✐❞❡♥t❡
❡♠ ❛❧❣✉♠ ✈ért✐❝❡ ❞❡ A′✱ ♦✉ s❡❥❛✱ ✉♠❛ ❝♦❜❡rt✉r❛ ♣♦r ✈ért✐❝❡s ❡♠ ¯
G✳
➱ ♣r♦✈❛❞♦ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐sã♦ s♦❜r❡ ❛ ✈✐❛❜✐❧✐❞❛❞❡ ❞❡ ❙❆▼❘❉P é N P✕ ❈♦♠♣❧❡t♦✱ ♣♦r ✉♠❛ r❡❞✉çã♦ ❛ ♣❛rt✐r ❞♦ ♣r♦❜❧❡♠❛ ❝❧áss✐❝♦ ❞❡ ❈♦❜❡rt✉r❛ ♣♦r ❱ért✐❝❡s✳ P❛r❛ ❙❆▼❘❉◆✱ sã♦ ❛♣r❡s❡♥t❛❞♦s s❡✐s ❛❧❣♦r✐t♠♦s ❝♦♠ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ 1/2 ♣❛r❛ ❝❧❛ss❡s ❡s♣❡❝✐❛✐s ❞❡ ❣r❛❢♦s ❞❡ r❡str✐çõ❡s✱ t❛❧ q✉❡ ♦ ❝á❧❝✉❧♦ ❞❡ ✉♠
❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ♠á①✐♠♦ ❡♠ G¯ s❡❥❛ ♣♦❧✐♥♦♠✐❛❧✳ ❉❡♥tr❡ ♦s s❡✐s ❛❧❣♦r✐t♠♦s✱ três
❞❡❧❡s s❡❣✉✐♥❞♦ ❛ ❛❜♦r❞❛❣❡♠ ❞❡♥♦♠✐♥❛❞❛ late❣❡r❛♠ s♦❧✉çõ❡s ❝♦♠ ♥♦ ♠í♥✐♠♦ ❛ ♠❡s♠❛
❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❛s ♦❜t✐❞❛s ❝♦♠ ♦s ❛❧❣♦r✐t♠♦s ❡q✉✐✈❛❧❡♥t❡s✱ s❡❣✉♥❞♦ ❛ ❛❜♦r❞❛❣❡♠early✳
➱ ♣r♦♣♦st♦ ✉♠ ♣ré✲♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♥stâ♥❝✐❛s ♣❛r❛ ❙❆▼❘❉◆ ❡ ❙❆▼❘❉P✳ ❚❡st❡s ❡①♣❡r✐♠❡♥t❛✐s s♦❜r❡ ♠♦❞❡❧♦s ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r ✐♥t❡✐r❛ ❞♦s ♣r♦❜❧❡♠❛s s✉❣❡✲ r❡♠ q✉❡✱ ❛♦ s❡ r❡❛❧✐③❛r ♦ ♣ré✲♣r♦❝❡ss❛♠❡♥t♦✱ s♦❧✉çõ❡s ♠❡❧❤♦r❡s sã♦ ♦❜t✐❞❛s✱ ♥♦ ♠❡s♠♦ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦✳ ➱ t❛♠❜é♠ ❛♣r❡s❡♥t❛❞♦ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ♠❡❧❤♦r✐❛ s♦❜r❡ ✉♠❛ s♦❧✉çã♦ ✐♥✐❝✐❛❧ ❞❡ ❙❆▼❘❉◆✱ ♦❜t✐❞❛ ❤❡✉r✐st✐❝❛♠❡♥t❡✱ q✉❡ é ❡♥tã♦ ✈✐♥❝✉❧❛❞♦ ❛ ✉♠❛ ♠❡✲ t❛❤❡✉ríst✐❝❛ ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❞❡ ❜✉s❝❛ ❧♦❝❛❧✳ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❡✈✐❞❡♥❝✐❛♠
q✉❡✱ ♣❛r❛ ✐♥stâ♥❝✐❛s ❡♠ q✉❡|V|>200✱ ❛ ❤❡✉ríst✐❝❛ é ♣r❡❢❡rí✈❡❧ à r❡s♦❧✉çã♦ ❞♦ ♠♦❞❡❧♦
❞❡ ♣r♦❣r❛♠❛çã♦ ✐♥t❡✐r❛ ♣♦r ✉♠ s♦❢t✇❛r❡ ❝♦♠❡r❝✐❛❧✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦✱ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✈❛s✱ ❆♥á❧✐s❡ ❈♦♠❜✐✲ ♥❛tór✐❛✱ ❆❧❣♦r✐t♠♦s✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ t❛❝❦❧❡ t❤❡ ▼❛①✐♠✉♠ ❆❝②❝❧✐❝ ❙✉❜❣r❛♣❤ ♣r♦❜❧❡♠ ✉♥❞❡r ◆❡❣❛t✐✈❡ ❉✐s✲ ❥✉♥❝t✐✈❡ ❈♦♥str❛✐♥ts ✭▼❆❙◆❉❈✮ ❛♥❞ t❤❡ ▼❛①✐♠✉♠ ❆❝②❝❧✐❝ ❙✉❜❣r❛♣❤ ♣r♦❜❧❡♠ ✉♥❞❡r P♦s✐t✐✈❡ ❉✐s❥✉♥❝t✐✈❡ ❈♦♥str❛✐♥ts ✭▼❆❙P❉❈✮✱ ❜♦t❤ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ N P✕❍❛r❞ ❝❧❛ss✳ ❉✐s❥✉♥❝t✐✈❡ ❝♦♥str❛✐♥ts ❝❛♥ ❜❡ ♥❡❣❛t✐✈❡✱ ✐♥ ✇❤✐❝❤ ❛ ❝❡rt❛✐♥ ♣❛✐r ♦❢ ❡♥t✐t✐❡s ❝❛♥♥♦t ❜❡ ❝♦♥t❛✐♥❡❞ s✐♠✉❧t❛♥❡♦✉s❧② ✐♥ ❛ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥❀ ♦r ♣♦s✐t✐✈❡ ✐♥ t❤❛t✱ ❣✐✈❡♥ ❛ ♣❛✐r ♦❢ ❡♥t✐t✐❡s✱ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ♠✉st ❝♦♠♣♦s❡ ❛ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥✳
■♥st❛♥❝❡s ♦❢ ▼❆❙◆❉❈ ❛♥❞ ▼❆❙P❉❈ ❛r❡ ❝♦♠♣♦s❡❞ ❜② ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ G = (V, A) ❛♥❞ ❛ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ G¯ = (A, E)✱ ✇❤✐❝❤ ❡♥❝♦❞❡s t❤❡ ❞✐s❥✉♥❝t✐✈❡ ♣♦s✐t✐✈❡ ♦r
♥❡❣❛t✐✈❡ ❝♦♥str❛✐♥ts ♦✈❡r ♣❛✐rs ♦❢ ❛r❝s ♦❢ ❣r❛♣❤G✳ ▼❆❙◆❉❈ ❝♦♥s✐sts ✐♥ ❞❡t❡r♠✐♥✐♥❣ ❛
♠❛①✐♠✉♠ s✉❜s❡t A′
⊆A ❢♦r ✇❤✐❝❤ t❤❡ s✉❜❣r❛♣❤ G′
= (V, A′
) ✐s ❝②❝❧❡ ❢r❡❡ ❛♥❞ A′ ✐s ❛
s❡t ♦❢ ✈❡rt✐❝❡s ✐♥ G¯ s✉❝❤ t❤❛t t❤❡r❡ ❛r❡ ♥♦ ❡❞❣❡s ❜❡t✇❡❡♥ t✇♦ ♦❢ ✐ts ✈❡rt✐❝❡s✱ ♥❛♠❡❧② ❛♥
✐♥❞❡♣❡♥❞❡♥t s❡t ❛t G¯✳ ▼❆❙P❉❈ ❝♦♥s✐sts ✐♥ ❞❡t❡r♠✐♥✐♥❣ ❛ ♠❛①✐♠✉♠ s✉❜s❡t A′
⊆ A
❢♦r ✇❤✐❝❤ t❤❡ s✉❜❣r❛♣❤ G′
= (V, A′
) ✐s ❝②❝❧❡ ❢r❡❡ ❛♥❞ A′ ✐s ❛ s❡t ♦❢ ✈❡rt✐❝❡s ✐♥ ¯
G s✉❝❤
t❤❛t ❡✈❡r② ❡❞❣❡ ✐♥ E ✐s ✐♥❝✐❞❡♥t ♦♥ s♦♠❡ ✈❡rt❡① ✐♥ A′✱ ♥❛♠❡❧② ❛ ✈❡rt❡① ❝♦✈❡r ✐♥ ¯
G✳
■t ✐s ♣r♦✈❡❞ t❤❛t t❤❡ ❢❡❛s✐❜✐❧✐t② ❞❡❝✐s✐♦♥ ✈❡rs✐♦♥ ♦❢ ▼❆❙P❉❈ ✐s N P✕❈♦♠♣❧❡t❡ ❜② ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❝❧❛ss✐❝ ❱❡rt❡① ❈♦✈❡r ♣r♦❜❧❡♠✳ ❋♦r ▼❆❙◆❉❈ ✇❡ ♣r♦♣♦s❡ s✐① ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✇✐t❤ ❝♦♥st❛♥t ❢❛❝t♦r ❡q✉❛❧ t♦ 1/2❢♦r s♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❣r❛♣❤
¯
G ✐♥ ✇❤✐❝❤ t❤❡ ♠❛①✐♠✉♠ ✐♥❞❡♣❡♥❞❡♥t s❡t ✐s ♣♦❧②♥♦♠✐❛❧✳ ❆♠♦♥❣ t❤❡ s✐① ❛❧❣♦r✐t❤♠s✱
t❤r❡❡ ♦❢ t❤❡♠ ❢♦❧❧♦✇✐♥❣ t❤❡ s♦✲❝❛❧❧❡❞late ❛♣♣r♦❛❝❤ ❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s ✇✐t❤ ❛t ❧❡❛st t❤❡
s❛♠❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ ♦♥❡s ♦❜t❛✐♥❡❞ ❜② ❡q✉✐✈❛❧❡♥t ❛❧❣♦r✐t❤♠s✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ early
❛♣♣r♦❛❝❤✳
❲❡ ♣r♦♣♦s❡ ❛ ♣r❡♣r♦❝❡ss✐♥❣ ♦❢ ▼❆❙P❉❈ ❛♥❞ ▼❆❙◆❉❈ ✐♥st❛♥❝❡s✳ ❈♦♠♣✉t❛t✐♦✲ ♥❛❧ ❡①♣❡r✐♠❡♥ts ♦♥ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♠♦❞❡❧s ♦❢ ❜♦t❤ ♣r♦❜❧❡♠s s✉❣❣❡st t❤❛t✱ ✇❤❡♥ ♣❡r❢♦r♠✐♥❣ t❤❡ ♣r❡♣r♦❝❡ss✐♥❣✱ s♦❧✉t✐♦♥s ♦❢ ❧❛r❣❡r ❝❛r❞✐♥❛❧✐t✐❡s ❛r❡ ♦❜t❛✐♥❡❞✱ ✐♥ t❤❡ s❛♠❡ r✉♥ t✐♠❡✳ ❲❡ ❛❧s♦ ♣r♦♣♦s❡❞ ❛♥ ✐♠♣r♦✈❡♠❡♥t ♣r♦❝❡❞✉r❡ ♦♥ ❛♥ ❤❡✉r✐st✐❝❛❧❧② ♦❜t❛✐♥❡❞ ✐♥✐t✐❛❧ s♦❧✉t✐♦♥ ♦❢ ▼❆❙◆❉❈✱ ✇❤✐❝❤ ✐s t❤❡♥ ❧✐♥❦❡❞ t♦ ❛ ♠❡t❛❤❡✉r✐st✐❝ ❛s ❛ ❧♦✲ ❝❛❧ s❡❛r❝❤ ♠❡t❤♦❞✳ ❈♦♠♣✉t❛t✐♦♥❛❧ r❡s✉❧ts s❤♦✇ t❤❛t ❢♦r ✐♥st❛♥❝❡s ✇✐t❤ |V|>200✱ t❤❡
❤❡✉r✐st✐❝ ✐s ♣r❡❢❡r❛❜❧❡ t♦ s♦❧✈✐♥❣ t❤❡ ✐♥t❡❣❡r ♣r♦❣r❛♠♠✐♥❣ ♠♦❞❡❧ ❜② ✉s✐♥❣ ❛ ❝♦♠♠❡r❝✐❛❧
s♦❢t✇❛r❡✳
❑❡②✇♦r❞s✿ ▼❛①✐♠✉♠ ❆❝②❝❧✐❝ ❙✉❜❣r❛♣❤✱ ❉✐s❥✉♥❝t✐✈❡ ❈♦♥str❛✐♥ts✱ ❈♦♠❜✐♥❛t♦r✐❛❧ ❆♥❛❧②s✐s✱ ❆❧❣♦r✐t❤♠s✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❊①❡♠♣❧♦ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❙❆▼✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ■♥stâ♥❝✐❛ ♣❛r❛ ❙❆▼❘❉◆ ❡ ❙❆▼❘❉P✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ❈♦♥❥✉♥t♦s ❞❡ ✈ért✐❝❡s A′ s❡❧❡❝✐♦♥❛❞♦s ❡♠ G¯✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✹ ❙♦❧✉çõ❡s ❞❛❞❛s ♣❡❧♦s ❣r❛❢♦s G′
= (V, A′
)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✸✳✶ ■♥stâ♥❝✐❛ ♣❛r❛ ❙❆▼❘❉P✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✷ ❙♦❧✉çã♦ ✐♥✈✐á✈❡❧ ♣❛r❛ ❙❆▼❘❉P✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✸ ●r❛❢♦ G= (V, E)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✸✳✹ ■♥stâ♥❝✐❛ ♣❛r❛ ❱❙❆❘❉P✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✺ ■♥stâ♥❝✐❛ ♣❛r❛ ❱❙❆❘❉P ❝♦♠ ❦❂✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✻ ■♥stâ♥❝✐❛ ♣❛r❛ ❱❙❆❘❉P ❝♦♠ ❦❂✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✼ ❙♦❧✉çã♦ ✐♥✈✐á✈❡❧ ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❞❡ ❱❙❆❘❉P ❝♦♠ ❦❂✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✽ ❖✉tr❛ s♦❧✉çã♦ ✐♥✈✐á✈❡❧ ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❞❡ ❱❙❆❘❉P ❝♦♠ ❦❂✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✹✳✶ ●r❛❢♦ G= (V, A)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✹✳✷ ■♥stâ♥❝✐❛ ♣❛r❛ ❙❆▼❘❉◆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✸ ●r❛❢♦ ♣❛r❛ ❡①❡♠♣❧✐✜❝❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ Degreel(G,G¯)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✹✳✹ ●r❛❢♦ G¯ = (A, E)❡ s✉❜❣r❛❢♦ ✐♥❞✉③✐❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✹✳✺ ❙♦❧✉çã♦ ót✐♠❛ ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❞❡ Degreel(G,G¯)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✺✳✶ ✭❚✶✮ ❛♣❧✐❝❛❞♦ ❛ ❙❆▼❘❉◆●✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✺✳✷ ✭❚✷✮ ❛♣❧✐❝❛❞♦ ❛ ❙❆▼❘❉◆●✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✳✸ ✭❚✸✮ ❛♣❧✐❝❛❞♦ ❛ ❙❆▼❘❉◆●✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✺✳✹ ✭❚✹✮ ❛♣❧✐❝❛❞♦ ❛ ❙❆▼❘❉◆●✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✺✳✺ ✭❚✺✮ ❛♣❧✐❝❛❞♦ ❛ ❙❆▼❘❉◆●✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✻✳✶ ❖r❞❡♥❛çã♦ t♦♣♦❧ó❣✐❝❛ ✐♥s❡r✐♥❞♦ ✈ért✐❝❡ vc✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✻✳✷ ●r❛✉ ♠é❞✐♦ ❞❡ ✈ért✐❝❡s ❡♠ G¯✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✹✳✶ ❈❛r❛❝t❡ríst✐❝❛s ❞❛s ✐♥stâ♥❝✐❛s t✐♣♦ T1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✹✳✷ ❘❡s✉❧t❛❞♦s ♣❛r❛ ❛❧❣♦r✐t♠♦s ❛♣r♦①✐♠❛t✐✈♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✸ ❊st❛tíst✐❝❛s s♦❜r❡ ❛s ♣♦r❝❡♥t❛❣❡♥s ❞❡ ❛r❝♦s ♥❛s s♦❧✉çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✹ ■♥t❡r✈❛❧♦s ❞❡ ❈♦♥✜❛♥ç❛ ✭✾✾✪✮ ♣❛r❛ ❛ ♠é❞✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✺ ❊st❛tíst✐❝❛s s♦❜r❡ ❛ ♠é❞✐❛ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❛s s♦❧✉çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✺✳✶ ❘❡s✉❧t❛❞♦s ❞❛ r❡❞✉çã♦ ❞❡ ❙❆▼❘❉◆● ♣❛r❛ T2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✺✳✷ ▼é❞✐❛s ♣❛r❛ r❡❞✉çã♦ ❡♠ T2 ♣❛r❛ ❙❆▼❘❉◆●✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✺✳✸ ❘❡s✉❧t❛❞♦s ❞❛ r❡❞✉çã♦ ❞❡ ❙❆▼❘❉P ♣❛r❛ T2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✺✳✹ ▼é❞✐❛s ♣❛r❛ r❡❞✉çã♦ ❡♠ T2 ♣❛r❛ ❙❆▼❘❉P✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✺✳✺ ❘❡s✉❧t❛❞♦s ❞❡ ♦t✐♠✐③❛çã♦ ♣❛r❛ ❙❆▼❘❉◆● s♦❜r❡ ✐♥stâ♥❝✐❛s T2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✺✳✻ ❈♦♠♣❛r❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ♦t✐♠✐③❛çã♦ ❞❡ ❙❆▼❘❉◆● ♣❛r❛ T2✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✺✳✼ ❘❡s✉❧t❛❞♦s ❞❡ ♦t✐♠✐③❛çã♦ ♣❛r❛ ❙❆▼❘❉P s♦❜r❡ ✐♥stâ♥❝✐❛s T2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✻✳✶ ❈♦♠♣❛r❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ❡①❛t♦s ❡ ❤❡✉ríst✐❝♦s ♣❛r❛ T2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
✻✳✷ ❘❡s✉❧t❛❞♦s ❞❡ ♦t✐♠✐③❛çã♦ ❳ ❤❡✉ríst✐❝♦s ♣❛r❛ ✐♥stâ♥❝✐❛s T2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
❙✉♠ár✐♦
❆❣r❛❞❡❝✐♠❡♥t♦s ✐①
❘❡s✉♠♦ ①✐✐✐
❆❜str❛❝t ①✈
▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✈✐✐
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✐①
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ❉❡✜♥✐çã♦ ❞♦s ♣r♦❜❧❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❈♦♠♣❧❡①✐❞❛❞❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❆❜♦r❞❛❣❡♥s ♣❛r❛ ♣r♦❜❧❡♠❛s N P✕❉✐❢í❝❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ ❖❜❥❡t✐✈♦s ❞♦ tr❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✺ ❖r❣❛♥✐③❛çã♦ ❞♦ tr❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✷ ❚r❛❜❛❧❤♦s ❘❡❧❛❝✐♦♥❛❞♦s ✶✶
✷✳✶ ❙❆▼ ❡ ♣r♦❜❧❡♠❛s ❡q✉✐✈❛❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✶ ❆❜♦r❞❛❣❡♥s ❡①❛t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✷ ❆❜♦r❞❛❣❡♥s ❤❡✉ríst✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✸ ❆❧❣♦r✐t♠♦s ❛♣r♦①✐♠❛t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✸ ❖ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✲
✈❛s P♦s✐t✐✈❛s ✶✼
✸✳✶ ❈♦♠♣❧❡①✐❞❛❞❡ ♣❛r❛ ♦ t❡st❡ ❞❡ ✈✐❛❜✐❧✐❞❛❞❡ ❞❡ ❙❆▼❘❉P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✶✳✶ ❈♦♥str✉çã♦ ❞❡ ✉♠❛ ✐♥stâ♥❝✐❛ ♣❛r❛ ❱❙❆❘❉P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✶✳✷ Pr♦✈❛ ❞❡ ◆P✲❈♦♠♣❧❡t✉❞❡ ♣❛r❛ ❱❙❆❘❉P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✷ ❋♦r♠✉❧❛çõ❡s ♣❛r❛ ❙❆▼❘❉P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✹ ❖ Pr♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✲
✈❛s ◆❡❣❛t✐✈❛s ✷✾
✹✳✶ ❆❧❣♦r✐t♠♦s ❛♣r♦①✐♠❛t✐✈♦s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ ✸✵ ✹✳✷ ❆❧❣♦r✐t♠♦s ❛♣r♦①✐♠❛t✐✈♦s ♣❛r❛ ❙❆▼❘❉◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✷✳✶ ❊①❡♠♣❧♦ ❞❡ ❡①❡❝✉çã♦ ❞♦ ❛❧❣♦r✐t♠♦ Degreel(G,G¯) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✹✳✸ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✺ Pré✲♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ■♥stâ♥❝✐❛s ✹✸
✺✳✶ ❘❡❞✉çã♦ ❞❡ ✐♥stâ♥❝✐❛s ♣❛r❛ ❙❆▼❘❉◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✶✳✶ ❚❡st❡s ❞❡ ❘❡❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✺✳✶✳✷ Pr♦❣r❛♠❛çã♦ ❧✐♥❡❛r ✐♥t❡✐r❛ ♣❛r❛ ❙❆▼❘❉◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✺✳✶✳✸ ■♠♣❧❡♠❡♥t❛çã♦ ♣❛r❛ ♣ré✲♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♥stâ♥❝✐❛s ❡♠
❙❆▼❘❉◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✺✳✶✳✹ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ♦ ♣ré✲♣r♦❝❡ss❛♠❡♥t♦ ❞❛s ✐♥stâ♥✲
❝✐❛s ❡♠ ❙❆▼❘❉◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✺✳✷ ❘❡❞✉çã♦ ❞❡ ✐♥stâ♥❝✐❛s ♣❛r❛ ❙❆▼❘❉P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✺✳✷✳✶ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ♦ ♣ré✲♣r♦❝❡ss❛♠❡♥t♦ ❞❛s ✐♥stâ♥✲ ❝✐❛s ❡♠ ❙❆▼❘❉P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✺✳✸ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ❛ ♦t✐♠✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✺✳✸✳✶ ❈♦♠♣❛r❛çã♦ ❞❛s s♦❧✉çõ❡s ❞❡ ❙❆▼❘❉◆● ❛♥t❡s ❡ ❛♣ós ♦ ♣ré✲ ♣r♦❝❡ss❛♠❡♥t♦ ❞❛ ✐♥stâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✺✳✸✳✷ ❈♦♠♣❛r❛çã♦ ❞❛s s♦❧✉çõ❡s ❞❡ ❙❆▼❘❉P ❛♥t❡s ❡ ❛♣ós ♦ ♣ré✲
♣r♦❝❡ss❛♠❡♥t♦ ❞❛ ✐♥stâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✻ ❍❡✉ríst✐❝❛ ❡ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ▼❡❧❤♦r✐❛ ♣❛r❛ ❙❆▼❘❉◆ ✻✶ ✻✳✶ ●❡r❛çã♦ ❞❡ ✉♠❛ s♦❧✉çã♦ ✐♥✐❝✐❛❧A′ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
✻✳✷ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ♠❡❧❤♦r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✻✳✷✳✶ ❖r❞❡♥❛çã♦ t♦♣♦❧ó❣✐❝❛ ❞♦s ✈ért✐❝❡s ❞❡ G ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
✻✳✸ ❍❡✉ríst✐❝❛ ■▲❙ ❛♣❧✐❝❛❞❛ ❛♦ ❙❆▼❘❉◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✻✳✸✳✶ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
✼ ❈♦♥❝❧✉sã♦ ✼✶
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✸
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
●r❛❢♦s sã♦ ❡str✉t✉r❛s ♠❛t❡♠át✐❝❛s ✉s❛❞❛s ♣❛r❛ ♠♦❞❡❧❛r r❡❧❛çõ❡s ❡♥tr❡ ♦❜❥❡t♦s ♦✉ s✐s✲ t❡♠❛s✱ ❡ ♣♦ss✉❡♠ ❛♣❧✐❝❛çõ❡s ❡♠ ❞✐❢❡r❡♥t❡s ❝❛♠♣♦s ❞❛ ❝✐ê♥❝✐❛✱ ❝♦♠♦ ♠❛t❡♠át✐❝❛✱ ❝♦♠✲ ♣✉t❛çã♦✱ ❡♥❣❡♥❤❛r✐❛✱ ❢ís✐❝❛✱ ❜✐♦❧♦❣✐❛✱ ❝✐ê♥❝✐❛s s♦❝✐❛✐s✱ ❡❝♦♥ô♠✐❝❛s ❡t❝✳ Pr♦❜❧❡♠❛s ❝♦♠ ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ♣♦❞❡♠ s❡r ♠♦❞❡❧❛❞❛s ❡♠ ❣r❛❢♦s✱ ❝♦♠♦ ♣r♦❜❧❡♠❛s ❞❡ ❡s❝❛❧♦♥❛♠❡♥t♦ ❞❡ t❛r❡❢❛s ❬❈♦✛♠❛♥ ❏r✳ ✫ ●r❛❤❛♠✱ ✶✾✼✷❪✱ r♦t❡❛♠❡♥t♦ ❬❘❡✐♥❡❧t✱ ✶✾✽✺❪✱ ❛♥á❧✐s❡ ❞❡ ✢✉①♦ ❞❡ ❞❛❞♦s ❬❘❛♠❛❝❤❛♥❞r❛♥✱ ✶✾✽✽❪ ❡t❝✳ ❆ s❡❣✉✐r✱ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❜ás✐❝❛s ✉s❛❞❛s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦ ❞❛ t❡s❡✳ P❛r❛ ♦✉tr❛s ❞❡✜♥✐çõ❡s ❡♠ t❡♦r✐❛ ❞♦s ❣r❛❢♦s✱ ✈✐❞❡ ❇♦♥❞② ✫ ▼✉rt② ❬✷✵✵✽❪❀ ❉✐❡st❡❧ ❬✷✵✵✺❪✳
❉❡✜♥✐çã♦ ✶✳ ●r❛❢♦ s✐♠♣❧❡s G= (V, E)✿ ❝♦♥s✐st❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ✈ért✐❝❡s V
❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s E✱ ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ♦r❞❡♥❛❞♦ ❞❡V ×V✳
❖s ❞♦✐s ✈ért✐❝❡s ❢♦r♠❛♥❞♦ ✉♠❛ ❛r❡st❛ sã♦ ❞✐t♦s s✉❛s ❡①tr❡♠✐❞❛❞❡s ❡ ❛ ❛r❡st❛ é ❞✐t❛ ✐♥❝✐❞❡♥t❡ ♣❛r❛ ❝♦♠ ♦s ✈ért✐❝❡s✳ ❯♠ ✈ért✐❝❡ u é ❞✐t♦ s❡r ❛❞❥❛❝❡♥t❡ ❛ ♦✉tr♦ ✈ért✐❝❡ v s❡
♦ ❣r❛❢♦ ❝♦♥t❡♠ ✉♠❛ ❛r❡st❛ {u, v}✳ ◆♦r♠❛❧♠❡♥t❡✱ q✉❛♥❞♦ s❡ tr❛t❛r ❞❡st❡ t✐♣♦ ❞❡ ❣r❛❢♦✱ ♦ ❛❞❥❡t✐✈♦ s✐♠♣❧❡s é ♦♠✐t✐❞♦✱ ❡st❛♥❞♦✱ ♥♦ ❡♥t❛♥t♦✱ s✉❜❡♥t❡♥❞✐❞♦✳
❉❡✜♥✐çã♦ ✷✳ ●r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦ G= (V, A)✿ ❝♦♥s✐st❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ✈ért✐❝❡s
V ❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛r❝♦s A✱ ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ♦r❞❡♥❛❞♦ ❞❡ V ×V✳
❯♠ ❛r❝♦ (v1, v2) é ✉♠ ❛r❝♦ q✉❡ s❛✐ ❞❡ v1 ❡ ❝❤❡❣❛ ❛ v2✳ ❖ ✈ért✐❝❡ v1 é ❞✐t♦ ♣r❡❞❡❝❡ss♦r
❞♦ ✈ért✐❝❡ v2 ❡ v2 é s✉❝❡ss♦r ❞❡ v1✱ ♦✉ v1 é ♦ ✈ért✐❝❡ ✐♥✐❝✐❛❧ ❡ v2 é ♦ ✈ért✐❝❡ ✜♥❛❧✳
❉❡✜♥✐çã♦ ✸✳ ●r❛❢♦ ❝♦♠♣❧❡t♦✿ ❣r❛❢♦ ♥♦ q✉❛❧ ❡①✐st❡ ✉♠❛ ❛r❡st❛ ❡♥tr❡ q✉❛✐sq✉❡r ❞♦✐s ✈ért✐❝❡s ❞✐st✐♥t♦s✳ ❯♠ ❣r❛❢♦ ❝♦♠♣❧❡t♦ ❝♦♠ n ✈ért✐❝❡s é ❞❡♥♦t❛❞♦ ♣♦r Kn✳
❉❡✜♥✐çã♦ ✹✳ ●r❛❢♦ ❜✐♣❛rt✐❞♦✿ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ♣♦❞❡ s❡r ♣❛rt✐❝✐♦♥❛❞♦ ❡♠ ❞♦✐s ❝♦♥❥✉♥t♦s✱ X ❡ Y✱ t❛❧ q✉❡ ❝❛❞❛ ❛r❡st❛ ♣♦ss✉✐ ✉♠ ❡①tr❡♠♦ ❡♠ X ❡ ♦ ♦✉tr♦ ❡♠ Y✳ ❖
♣❛r (X, Y) é ❞✐t♦ ✉♠❛ ❜✐♣❛rt✐çã♦ ❞♦ ❣r❛❢♦✳
✷ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦
❉❡✜♥✐çã♦ ✺✳ ●r❛❢♦ ❜✐♣❛rt✐❞♦ ❝♦♠♣❧❡t♦✿ ❣r❛❢♦ ❝♦♠ ❜✐♣❛rt✐çã♦ (X, Y)✱ t❛❧ q✉❡ ❝❛❞❛
✈ért✐❝❡ ❞❡X ❡stá ❧✐❣❛❞♦ ❛ t♦❞♦s ♦s ✈ért✐❝❡s ❞❡Y✳ ❙❡|X|=m ❡ |Y|=n✱ ❡♥tã♦ ♦ ❣r❛❢♦
❜✐♣❛rt✐❞♦ ❝♦♠♣❧❡t♦ é ❞❡♥♦t❛❞♦ ♣♦r Km,n✳
❉❡✜♥✐çã♦ ✻✳ ❙✉❜❣r❛❢♦✿ ✉♠ ❣r❛❢♦ H é s✉❜❣r❛❢♦ ❞❡ G✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r H ⊆ G✱ s❡ V(H)⊆V(G) ❡ E(H)⊆E(G)✳
❉❡✜♥✐çã♦ ✼✳ ❙✉❜❣r❛❢♦ ✐♥❞✉③✐❞♦ ♣♦r ✈ért✐❝❡s✿ s❡❥❛ V′ ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ V✳
❖ s✉❜❣r❛❢♦ ❞❡ G ❝✉❥♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s é V′ ❡ ❝✉❥♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s é ❝♦♠♣♦st♦
♣❡❧❛s ❛r❡st❛s ❞❡Gt❛❧ q✉❡ ❛s ❞✉❛s ❡①tr❡♠✐❞❛❞❡s ♣❡rt❡♥ç❛♠ ❛V′✱ é ✉♠ s✉❜❣r❛❢♦ ✐♥❞✉③✐❞♦
♣❡❧♦s ✈ért✐❝❡s V′✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r
G[V′
]✳
❉❡✜♥✐çã♦ ✽✳ ❙✉❜❣r❛❢♦ ✐♥❞✉③✐❞♦ ♣♦r ❛r❡st❛s✿ s❡❥❛ E′ ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡
E✳
❖ s✉❜❣r❛❢♦ ❞❡G❝✉❥♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡①tr❡♠♦s ❞❛s ❛r❡st❛s ❡♠E′✱
❡ ❝✉❥♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s éE′✱ é ✉♠ s✉❜❣r❛❢♦ ✐♥❞✉③✐❞♦ ♣❡❧❛s ❛r❡st❛s
E′✱ r❡♣r❡s❡♥t❛❞♦
♣♦r G[E′
]✳
❉❡✜♥✐çã♦ ✾✳ ●r❛✉ ❞❡ ✉♠ ✈ért✐❝❡✿ ♦ ❣r❛✉ ❞❡ ✉♠ ✈ért✐❝❡✱δ(v)✱ ❡♠ ✉♠ ❣r❛❢♦ é ♦ ♥ú♠❡r♦
❞❡ ❛r❡st❛s ✐♥❝✐❞❡♥t❡s ❛ ❡st❡ ✈ért✐❝❡ ♥♦ ❣r❛❢♦✳ ❙❡ ♦ ❣r❛❢♦ é ❞✐r❡❝✐♦♥❛❞♦✱ ❤á q✉❡ s❡ ❞✐st✐♥❣✉✐r ❡♥tr❡ ♦ ❣r❛✉ ❞❡ ❡♥tr❛❞❛ ❞♦ ✈ért✐❝❡ δ+(v)✱ ❞❛❞♦ ♣❡❧♦ ♥ú♠❡r♦ ❞❡ ❛r❝♦s q✉❡
♣♦ss✉❡♠ v ❝♦♠♦ ✈ért✐❝❡ ✜♥❛❧✱ ❡ ♦ ❣r❛✉ ❞❡ s❛í❞❛ ❞♦ ✈ért✐❝❡ δ−
(v)✱ ❞❛❞♦ ♣❡❧♦ ♥ú♠❡r♦ ❞❡
❛r❝♦s q✉❡ ♣♦ss✉❡♠ v ❝♦♠♦ ✈ért✐❝❡ ✐♥✐❝✐❛❧✳
❉❡✜♥✐çã♦ ✶✵✳ ❈❛♠✐♥❤♦✿ ✉♠ ❝❛♠✐♥❤♦ ❡♠ ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦G é ❞❡✜♥✐❞♦ ♣♦r ✉♠❛
s❡q✉ê♥❝✐❛ ✜♥✐t❛✱ ♥ã♦ ♥✉❧❛✱ W = v0a1v1a2v2. . . akvk✱ ❝✉❥♦s t❡r♠♦s s❡ ❛❧t❡r♥❛♠ ❡♥tr❡
✈ért✐❝❡s ❡ ❛r❝♦s t❛❧ q✉❡✱ ♣❛r❛ 0 < i ≤ k✱ ai t❡♠ ❝♦♠♦ ✈ért✐❝❡ ✐♥✐❝✐❛❧ vi−1 ❡ ✈ért✐❝❡
✜♥❛❧ vi✳ W é ✉♠ ❝❛♠✐♥❤♦ ❞❡ v0 ❛ vk s❡ ❡❧❡ s❡ ✐♥✐❝✐❛ ❡♠ v0 ❡ t❡r♠✐♥❛ ❡♠ vk✱ ❡ ♣♦❞❡
s❡r ❞❡♥♦t❛❞♦ ♣♦r v0 ❀ vk✳ ❆❧é♠ ❞✐st♦✱ ❡♠ ✉♠ ❣r❛❢♦ s✐♠♣❧❡s✱ ✉♠ ❝❛♠✐♥❤♦ ♣♦❞❡ s❡r
❡s♣❡❝✐✜❝❛❞♦ s✐♠♣❧❡s♠❡♥t❡ ♣❡❧❛ s❡q✉ê♥❝✐❛ ❞❡ s❡✉s ✈ért✐❝❡s v0v1v2. . . vk✳
❉❡✜♥✐çã♦ ✶✶✳ ❈❛♠✐♥❤♦ s✐♠♣❧❡s✿ ❝❛❞❛ ✈ért✐❝❡ ❞♦ ❝❛♠✐♥❤♦ ❛♣❛r❡❝❡ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③✳
❉❡✜♥✐çã♦ ✶✷✳ ❈♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❝❛♠✐♥❤♦✿ é ❛ s♦♠❛ ❞♦ ♥ú♠❡r♦ ❞❡ ❛r❝♦s q✉❡ ♦ ❝♦♠♣õ❡✳
❉❡✜♥✐çã♦ ✶✸✳ ❈✐❝❧♦✿ ❝❛♠✐♥❤♦ C = v0a1v1a2v2. . . akvkak+1v0✱ ❞❡♥♦♠✐♥❛❞♦ ❝❛♠✐♥❤♦
✶✳✶✳ ❉❡❢✐♥✐çã♦ ❞♦s ♣r♦❜❧❡♠❛s ✸
❉❡✜♥✐çã♦ ✶✹✳ ➪r✈♦r❡✿ ❣r❛❢♦ ♥ã♦ ❞✐r❡❝✐♦♥❛❞♦ ❡♠ q✉❡ q✉❛✐sq✉❡r ❞♦✐s ✈ért✐❝❡s ❡stã♦ ❝♦♥❡❝t❛❞♦s ♣♦r ❡①❛t❛♠❡♥t❡ ✉♠ ú♥✐❝♦ ❝❛♠✐♥❤♦ s✐♠♣❧❡s✳
❉❡✜♥✐çã♦ ✶✺✳ ➪r✈♦r❡ ❜✐♥ár✐❛✿ ♣♦❞❡ s❡r ❞❡✜♥✐❞❛ ❝♦♠♦ ✉♠❛ ár✈♦r❡ ❡♥r❛✐③❛❞❛ ❡♠ q✉❡ ❝❛❞❛ ✈ért✐❝❡ tê♠ ♥♦ ♠á①✐♠♦ ❞♦✐s ✜❧❤♦s✱ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦s ✈ért✐❝❡s ❛❞❥❛❝❡♥t❡s ❛♦ ✈ért✐❝❡ ❞❡ ♦r✐❣❡♠✱ ❞❡♥♦♠✐♥❛❞♦ ♣❛✐✳ ❯♠ ✈ért✐❝❡ ❢♦❧❤❛ é ✉♠ ✈ért✐❝❡ q✉❡ ♥ã♦ ♣♦ss✉✐ ✜❧❤♦s✳ ❖ ♥í✈❡❧ ❞❡ ✉♠ ✈ért✐❝❡ é ❛ s✉❛ ❞✐stâ♥❝✐❛ à r❛✐③✳
❉❡✜♥✐çã♦ ✶✻✳ ❈♦♠♣♦♥❡♥t❡ ❢♦rt❡♠❡♥t❡ ❝♦♥❡①❛✿ ❞❛❞♦ ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦✱ ❞♦✐s ✈ér✲ t✐❝❡s u ❡ v sã♦ ❞✐t♦s ❝♦♥❡❝t❛❞♦s s❡ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ❞❡ u ♣❛r❛ v ❡ ❞❡ v ♣❛r❛ u ❡♠ G✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ❝♦♥❡❝t❛❞♦s ❞❡✜♥❡♠ ✉♠❛ ♣❛rt✐çã♦ ❞❡ V ❡♠ s✉❜❝♦♥❥✉♥t♦s V1, V2, . . . , Vω✱ t❛❧ q✉❡ ❞♦✐s ✈ért✐❝❡s u ❡ v ❡stã♦ ❝♦♥❡❝t❛❞♦s s❡ ❡ s♦♠❡♥t❡ s❡ ❛♠❜♦s ♣❡r✲
t❡♥❝❡♠ ❛♦ ♠❡s♠♦ s✉❜❝♦♥❥✉♥t♦ Vi✳ ❖s s✉❜❣r❛❢♦s ✐♥❞✉③✐❞♦s G[V1], G[V2], . . . G[Vω] sã♦
❝❤❛♠❛❞♦s ❝♦♠♣♦♥❡♥t❡s ❢♦rt❡♠❡♥t❡ ❝♦♥❡①❛s ❞❡ G✳
❉❡✜♥✐çã♦ ✶✼✳ ●r❛❢♦ ❢♦rt❡♠❡♥t❡ ❝♦♥❡①♦✿ ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦ G é ❞✐t♦ ❢♦rt❡♠❡♥t❡
❝♦♥❡①♦ s❡ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ ❝♦♠♣♦♥❡♥t❡ ❢♦rt❡♠❡♥t❡ ❝♦♥❡①❛✱ ♦✉ s❡❥❛✱ s❡ ❝♦♥té♠ ✉♠ ❝❛♠✐♥❤♦ ❞✐r❡❝✐♦♥❛❞♦ ❞❡ u ♣❛r❛ v ❡ ❞❡ v ♣❛r❛u ♣❛r❛ ❝❛❞❛ ♣❛r ❞❡ ✈ért✐❝❡s u, v✳
❉❡✜♥✐çã♦ ✶✽✳ ❈❧✐q✉❡✿ ❞❛❞♦ ✉♠ ❣r❛❢♦ G = (V, E)✱ ✉♠❛ ❝❧✐q✉❡ é ✉♠ s✉❜❝♦♥❥✉♥t♦
V′ ⊆V t❛❧ q✉❡ t♦❞❛ ❞✉♣❧❛ ❞❡ ✈ért✐❝❡s ❡♠ V′ ❡stá ❧✐❣❛❞❛ ♣♦r ✉♠❛ ❛r❡st❛ ❡♠ E✳
❉❡✜♥✐çã♦ ✶✾✳ ❈♦❜❡rt✉r❛ ♣♦r ✈ért✐❝❡s✿ ❞❛❞♦ ✉♠ ❣r❛❢♦ G= (V, E)✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡
✈ért✐❝❡s V′ ⊆
V s❡rá ✉♠❛ ❝♦❜❡rt✉r❛ s❡ t♦❞❛ ❛r❡st❛ ❞❡E t❡♠ ✉♠ ❡①tr❡♠♦ ❡♠ ✉♠ ✈ért✐❝❡
❞❡ V′✳
❉❡✜♥✐çã♦ ✷✵✳ ❈♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡✿ ❞❛❞♦ ✉♠ ❣r❛❢♦G= (V, E)✱ t❡♠✲s❡ ✉♠ ❝♦♥❥✉♥t♦
✐♥❞❡♣❡♥❞❡♥t❡ I ⊆V s❡ ♥❡♥❤✉♠ ✈ért✐❝❡ ❞❡ I é ❛❞❥❛❝❡♥t❡ ❛ ♦✉tr♦ ✈ért✐❝❡ ❞❡ I ♦✉✱ ♣❛r❛
t♦❞♦ u, v ∈ I✱ ❛ ❛r❡st❛ {u, v} ∈/ E✳ ❖ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ é
✉♠❛ ❝♦❜❡rt✉r❛ ♣♦r ✈ért✐❝❡s✳
✶✳✶ ❉❡✜♥✐çã♦ ❞♦s ♣r♦❜❧❡♠❛s
✹ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦
ú♥✐❝❛ ♣♦♣✉❧❛çã♦ ❞❡ ❝❛❞❛ ❡s♣é❝✐❡✱ ❡①✐st❡♠ t❛♠❜é♠ ♠❛♣❛s ❝r✐❛❞♦s ❛ ♣❛rt✐r ❞❡ ♠ú❧t✐♣❧❛s ♣♦♣✉❧❛çõ❡s✳ ❊♠ ❢❛❝❡ ❞❡st❡s ♠❛♣❛s✱ s✉r❣✐✉ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❡♥❝♦♥tr❛r ✉♠ ♠❛♣❛ ❝♦♥s❡♥s✉❛❧✱ ♦✉ s❡❥❛✱ ✉♠ ♠❛♣❛ q✉❡ ❝♦♠❜✐♥❛ ❛s ✐♥❢♦r♠❛çõ❡s ❞❡ ✈ár✐♦s ♠❛♣❛s ❣❡♥ét✐❝♦s ❞❡ ♦r❞❡♥s ♣❛r❝✐❛✐s ✭♠♦❞❡❧❛❞♦s ♣♦r ❣r❛❢♦s ❛❝í❝❧✐❝♦s ❞✐r❡❝✐♦♥❛❞♦s✮ ❡ ♣♦ss✐✈❡❧♠❡♥t❡ ✐♥❝♦♥✲ s✐st❡♥t❡s✳ ❙❡♥❞♦ ❛ss✐♠✱ ♦s ❛✉t♦r❡s ♠♦❞❡❧❛r❛♠ ❝❛❞❛ ♠❛♣❛ ❞❡ ❡♥tr❛❞❛ ❝♦♠♦ ✉♠❛ ♦r❞❡♠ ♣❛r❝✐❛❧ ❡ ❢♦r♠✉❧❛r❛♠ ♦ ♣r♦❜❧❡♠❛ ❞♦ ♠❛♣❛ ❣❡♥ét✐❝♦ ❝♦♥s❡♥s✉❛❧ ❝♦♠♦ ❡♥❝♦♥tr❛r ✉♠❛ ♦r✲ ❞❡♠ ♣❛r❝✐❛❧ ♠❡❞✐❛♥❛ ❞♦s ♠❛r❝❛❞♦r❡s ❣❡♥ét✐❝♦s✳ ❆ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❡♥✈♦❧✈❡ ❡♥tã♦ ❡♥❝♦♥tr❛r ♦ ♠á①✐♠♦ s✉❜❣r❛❢♦ ❛❝í❝❧✐❝♦ ❞❡ ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦ ♣♦♥❞❡r❛❞♦✱ ❡♠ q✉❡ ♦s ✈ért✐❝❡s ❞♦ ❣r❛❢♦ ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ♠❛r❝❛❞♦r❡s ❣❡♥ét✐❝♦s ❡♠ ✉♠ ♠❛♣❛ ❡ ♦s ❝❛♠✐♥❤♦s ❝♦rr❡s♣♦♥❞❡♠ às ✐♥❢♦r♠❛çõ❡s ❞❡ ♦r❞❡♥❛çã♦✳
❖ ♣r♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ ✭❞♦ ✐♥❣❧ês✱ ▼❛①✐♠✉♠ ❆❝②❝❧✐❝ ❙✉❜✲ ❣r❛♣❤✮✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❙❆▼✱ ❝♦♥s✐st❡ ❡♠✱ ❞❛❞♦ ✉♠ ❣r❛❢♦ G = (V, A)✱ ❞❡t❡r♠✐♥❛r
✉♠ s✉❜❝♦♥❥✉♥t♦ ♠á①✐♠♦ A′
⊆ A ♣❛r❛ ♦ q✉❛❧ ♦ s✉❜❣r❛❢♦ G′
= (V, A′
) s❡❥❛ ❛❝í❝❧✐❝♦✳ ❆
❋✐❣✉r❛ ✶✳✶✭❛✮ ❛♣r❡s❡♥t❛ ✉♠❛ ✐♥stâ♥❝✐❛ ♣❛r❛ ♦ ❙❆▼ ❡ ❛ ❋✐❣✉r❛ ✶✳✶✭❜✮ ❛♣r❡s❡♥t❛ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❞❛❞❛✳ ◆♦t❡ q✉❡ ♣❛r❛ ❡st❛ ✐♥stâ♥❝✐❛✱ ♦ ❣r❛❢♦G♣♦ss✉✐ ❞♦✐s ❝✐❝❧♦s
❞✐s❥✉♥t♦s✱ c1 = (v1a1v2a2v3a3v1) ❡ c2 = (v4a4v5a5v6a6v4)✱ ♣♦rt❛♥t♦ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ❞❡
s❡✉s ❛r❝♦s✱ ❝❛❞❛ ✉♠ ♣❡rt❡♥❝❡♥t❡ ❛ ✉♠ ❞♦s ❝✐❝❧♦s✱ ♥ã♦ ♣♦❞❡rã♦ ❝♦♠♣♦r ♦ ❝♦♥❥✉♥t♦ A′✳
✭❛✮ ●r❛❢♦G= (V, A)✳ ✭❜✮ ●r❛❢♦G′= (V, A′)✳
❋✐❣✉r❛ ✶✳✶✳ ❊①❡♠♣❧♦ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❙❆▼✳
❆❧é♠ ❞❡ ♣r♦❜❧❡♠❛s ❝❧áss✐❝♦s q✉❡ ♣♦❞❡♠ s❡r ❢♦r♠✉❧❛❞♦s ❡♠ ❣r❛❢♦s✱ ♣♦❞❡✲s❡ ❝✐✲ t❛r ❣❡♥❡r❛❧✐③❛çõ❡s ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ♣♦ss✉❡♠ ❝❡rt❛s r❡str✐çõ❡s✳ ❈❤❛♠❛♠✲s❡ r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s ❜✐♥ár✐❛s ❛q✉❡❧❛s ♥❛s q✉❛✐s ✉♠ ❝❡♥ár✐♦ ♦✉ ♦✉tr♦ ♣♦❞❡ ♦❝♦rr❡r✳ ❘❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s ❜✐♥ár✐❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s s♦❜ ✈ár✐❛s ❝❧❛ss✐✜❝❛çõ❡s✱ ❞❡♥tr❡ ❡❧❛s✿
✶✳✶✳ ❉❡❢✐♥✐çã♦ ❞♦s ♣r♦❜❧❡♠❛s ✺
✷✳ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✈❛s P♦s✐t✐✈❛s ♦✉ ❞❡ ❈♦❛çã♦✿ ♣❛r❛ ❝❛❞❛ ♣❛r ❞❡ ❡♥t✐❞❛❞❡s✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❡❧❛s ❞❡✈❡ ❝♦♠♣♦r ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧✳
❱ár✐♦s ♣r♦❜❧❡♠❛s ❝❧áss✐❝♦s ❡♠ t❡♦r✐❛ ❞♦s ❣r❛❢♦s ❢♦r♠✉❧❛❞♦s s♦❜ r❡str✐çõ❡s ❞✐s❥✉♥✲ t✐✈❛s✱ ♣♦s✐t✐✈❛s ♦✉ ♥❡❣❛t✐✈❛s✱ ❝♦♠♦ ♦s ♣r♦❜❧❡♠❛s ❞❡ ➪r✈♦r❡ ●❡r❛❞♦r❛ ▼í♥✐♠❛✱ ❊♠♣❛✲ r❡❧❤❛♠❡♥t♦ ▼á①✐♠♦✱ ❈❛♠✐♥❤♦ ▼í♥✐♠♦ ❡ ❋❧✉①♦ ▼á①✐♠♦✱ ❡♥❝♦♥tr❛♠✲s❡ ❞✐s♣♦♥í✈❡✐s ♥❛ ❧✐t❡r❛t✉r❛ ❬❑❛♥♥✱ ✶✾✾✸❀ ❉❛r♠❛♥♥ ❡t ❛❧✳✱ ✷✵✵✾✱ ✷✵✶✶❀ P❢❡rs❝❤② ✫ ❙❝❤❛✉❡r✱ ✷✵✶✸❀ ❩❤❛♥❣ ❡t ❛❧✳✱ ✷✵✶✶❪✳
◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ s❡rã♦ ❛❜♦r❞❛❞♦s ♦s ♣r♦❜❧❡♠❛s ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✈❛s P♦s✐t✐✈❛s ✭❙❆▼❘❉P✮ ❡ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡s✲ tr✐çõ❡s ❉✐s❥✉♥t✐✈❛s ◆❡❣❛t✐✈❛s ✭❙❆▼❘❉◆✮✱ ♥♦s q✉❛✐s ❛s r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s ✐♥❝✐❞❡♠ s♦❜r❡ ♣❛r❡s ❞❡ ❛r❝♦s ❞♦ ❣r❛❢♦ G= (V, A)✳ ➱ ❝♦♥✈❡♥✐❡♥t❡ r❡♣r❡s❡♥t❛r ❛s r❡str✐çõ❡s ❞✐s✲
❥✉♥t✐✈❛s s♦❜ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛r❝♦s ❡♠ t❡r♠♦s ❞❡ ✉♠ ❣r❛❢♦ ♥ã♦ ❞✐r❡❝✐♦♥❛❞♦G¯ = (A, E)✱
❞❡♥♦♠✐♥❛❞♦ ❣r❛❢♦ ❞❡ ❝♦♥✢✐t♦s ♣❛r❛ ❛s r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s ♥❡❣❛t✐✈❛s ♦✉ ❣r❛❢♦ ❞❡ ❝♦✲ ❛çã♦ ♣❛r❛ ❛s r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s ♣♦s✐t✐✈❛s✳ ◆❡st❡ ❣r❛❢♦✱ ♦s ✈ért✐❝❡s ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ❛r❝♦s ❞♦ ❣r❛❢♦ G ❡ ❛s ❛r❡st❛s r❡♣r❡s❡♥t❛♠ ❛s r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s✳ ❙❡❣✉❡♠ ❛s
❞❡✜♥✐çõ❡s ❢♦r♠❛✐s ❞♦s ❞♦✐s ♣r♦❜❧❡♠❛s✳
❉❡✜♥✐çã♦ ✷✶✳ ❙❆▼❘❉P✿ s❡❥❛ G = (V, A) ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦ ❡ G¯ = (A, E) ♦
❣r❛❢♦ ❞❡ ❝♦❛çã♦ ❛ss♦❝✐❛❞♦✳ ❖ ♣r♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✈❛s P♦s✐t✐✈❛s ❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r ✉♠ s✉❜❝♦♥❥✉♥t♦ A′ ⊆A ❞❡ ❝❛r❞✐♥❛❧✐❞❛❞❡
♠á①✐♠❛✱ ♥♦ q✉❛❧ ♦ s✉❜❣r❛❢♦ G′
= (V, A′
) s❡❥❛ ❛❝í❝❧✐❝♦ ❡ A′ s❡❥❛ ✉♠❛ ❝♦❜❡rt✉r❛ ♣♦r
✈ért✐❝❡s ❡♠ G¯✳
❉❡✜♥✐çã♦ ✷✷✳ ❙❆▼❘❉◆✿ s❡❥❛ G= (V, A) ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦ ❡ G¯ = (A, E) ♦ ❣r❛❢♦
❞❡ ❝♦♥✢✐t♦s ❛ss♦❝✐❛❞♦✳ ❖ ♣r♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡str✐çõ❡s ❉✐s✲ ❥✉♥t✐✈❛s ◆❡❣❛t✐✈❛s ❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r ✉♠ s✉❜❝♦♥❥✉♥t♦ A′ ⊆
A ❞❡ ❝❛r❞✐♥❛❧✐❞❛❞❡
♠á①✐♠❛✱ ♥♦ q✉❛❧ ♦ s✉❜❣r❛❢♦ G′
= (V, A′
) s❡❥❛ ❛❝í❝❧✐❝♦ ❡ A′ s❡❥❛ ✉♠ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥✲
❞❡♥t❡ ❡♠ G¯✳
❆ ❋✐❣✉r❛ ✶✳✷ ❛♣r❡s❡♥t❛ ✉♠❛ ✐♥stâ♥❝✐❛ ♣❛r❛ ❡①❡♠♣❧✐✜❝❛r ❛ ❛♣❧✐❝❛çã♦ ❞❡ ❙❆▼❘❉◆ ❡ ❙❆▼❘❉P✳ ◆♦ ❣r❛❢♦ G✭❋✐❣✉r❛ ✶✳✷✭❛✮✮✱ ❡①✐st❡ ✉♠❛ r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛✱ s❡❥❛ ♥❡❣❛t✐✈❛
♦✉ ♣♦s✐t✐✈❛✱ s♦❜r❡ ♦ ♣❛r ❞❡ ❛r❝♦s ④✭✷✱✶✮✱✭✹✱✶✮⑥✱ ♣♦r ✐st♦ ❡st❡s ❛r❝♦s ❡stã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r tr❛❝❡❥❛❞♦s✳ ◆♦t❡ q✉❡ ❡st❡s ❛r❝♦s sã♦ ✈ért✐❝❡s ♥♦ ❣r❛❢♦G¯ ✭❋✐❣✉r❛ ✶✳✷✭❜✮✮✱ ✉♥✐❞♦s ♣♦r
✉♠❛ ❛r❡st❛✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ r❡str✐çã♦ ❞✐s❥✉♥t✐✈❛✳ ❆ ❋✐❣✉r❛ ✶✳✸ ❛♣r❡s❡♥t❛ ♦s ❣r❛❢♦s G¯ ❡
♦s ❝♦♥❥✉♥t♦s ❞❡ ✈ért✐❝❡s A′✱ ❞❡st❛❝❛❞♦s ♥❛ ❝♦r ♣r❡t❛✱ s❡❧❡❝✐♦♥❛❞♦s ♣❛r❛ ♣❡r❢❛③❡r❡♠ ✉♠
✻ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦
❡♠G✳ P♦r ✐st♦✱ ♦ ✈ért✐❝❡(2,3)∈/ A′✱ ♣❛r❛ ❛♠❜♦s ♦s ♣r♦❜❧❡♠❛s✱ ❞❡s❢❛③❡♥❞♦ ♦ ❝✐❝❧♦ ❡♠G✳
❆ ❋✐❣✉r❛ ✶✳✹ ❛♣r❡s❡♥t❛ ❛s s♦❧✉çõ❡s ♣❛r❛ ♦s ♣r♦❜❧❡♠❛s✱ ❞❛❞❛s ♣❡❧♦s ❣r❛❢♦sG′
= (V, A′
)✳
✭❛✮ ●r❛❢♦G= (V, A)✳ ✭❜✮ ●r❛❢♦G¯= (A, E)✳
❋✐❣✉r❛ ✶✳✷✳ ■♥stâ♥❝✐❛ ♣❛r❛ ❙❆▼❘❉◆ ❡ ❙❆▼❘❉P✳
✭❛✮ A′ ♣❛r❛ ❙❆▼❘❉◆✳ ✭❜✮A′ ♣❛r❛ ❙❆▼❘❉P✳
❋✐❣✉r❛ ✶✳✸✳ ❈♦♥❥✉♥t♦s ❞❡ ✈ért✐❝❡sA′ s❡❧❡❝✐♦♥❛❞♦s ❡♠ G¯✳
✭❛✮ G′= (V, A′)♣❛r❛ ❙❆▼❘❉◆✳ ✭❜✮ G′ = (V, A′)♣❛r❛ ❙❆▼❘❉P✳
❋✐❣✉r❛ ✶✳✹✳ ❙♦❧✉çõ❡s ❞❛❞❛s ♣❡❧♦s ❣r❛❢♦s G′
= (V, A′
)✳
✶✳✷✳ ❈♦♠♣❧❡①✐❞❛❞❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ✼
♣♦ssí✈❡❧ ❞❡ ♣r❡❢❡rê♥❝✐❛s✳ ❘❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s ♣♦s✐t✐✈❛s ♣♦❞❡♠ ❝♦♥st✐t✉✐r ✉♠❛ ❢♦r♠❛ ❞❡ ❢♦rç❛r ♣r❡❢❡rê♥❝✐❛s ❝♦♥s✐❞❡r❛❞❛s ✐♠♣♦rt❛♥t❡s ♥♦ ♣r♦❥❡t♦✳
❈♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ ♣❡sq✉✐s❛✱ ❢♦r❛♠ ✐♥tr♦❞✉③✐❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ♦s ♣r♦✲ ❜❧❡♠❛s ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✈❛s ◆❡❣❛t✐✈❛s ❡ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✈❛s P♦s✐t✐✈❛s✱ ♠♦❞❡❧❛❞♦s ♣♦r ❣r❛❢♦s ❞❡ ❝♦♥✢✐t♦s ❡ ❝♦❛çã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❬▼❛♣❛ ✫ ❯rr✉t✐❛✱ ✷✵✶✺❪✳ ❆té ♦ ♣r❡s❡♥t❡ ♠♦♠❡♥t♦✱ ♥ã♦ s❡ ❝♦✲ ♥❤❡❝❡♠ ❞✐s♣♦♥í✈❡✐s ♥❛ ❧✐t❡r❛t✉r❛ ❛♣❧✐❝❛çõ❡s ❞✐r❡t❛s ♣❛r❛ ❙❆▼❘❉◆ ♦✉ ❙❆▼❘❉P✱ ♠❛s t❛♥t♦ ♦ ❙❆▼ q✉❛♥t♦ ❛s r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s sã♦ ✐♠♣♦rt❛♥t❡s ♥❛ ♣rát✐❝❛✳ ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ✈✐s❛ ❢♦❝❛r ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ t❡ór✐❝♦✱ s❡♠ ♠❛✐s ❞✐s❝✉rs❛r s♦❜r❡ ♣♦ssí✈❡✐s ❛♣❧✐❝❛çõ❡s✳
✶✳✷ ❈♦♠♣❧❡①✐❞❛❞❡ ❝♦♠♣✉t❛❝✐♦♥❛❧
❖s três ♣r♦❜❧❡♠❛s ❛♣r❡s❡♥t❛❞♦s ♣♦r ❉❛r♠❛♥♥ ❡t ❛❧✳ ❬✷✵✶✶❪✱ ❝❛♠✐♥❤♦s ♠❛✐s ❝✉rt♦s✱ ár✈♦r❡s ❣❡r❛❞♦r❛s ♠í♥✐♠❛s ❡ ❡♠♣❛r❡❧❤❛♠❡♥t♦s ♠á①✐♠♦s✱ sã♦ r❡s♦❧✈✐❞♦s ❡♠ t❡♠♣♦ ♣♦✲ ❧✐♥♦♠✐❛❧ ❡♠ s✉❛s ❢♦r♠❛s ❝❧áss✐❝❛s ❡ s❡ t♦r♥❛♠ N P✕❉✐❢í❝❡✐s q✉❛♥❞♦ s✉❜♠❡t✐❞♦s às r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s✳ ◗✉❛♥t♦ à s✉❛ ❝♦♠♣❧❡①✐❞❛❞❡✱ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ éN P✕❉✐❢í❝✐❧ ❬❑❛r♣✱ ✶✾✼✷❀ ●❛r❡② ✫ ❏♦❤♥s♦♥✱ ✶✾✼✾❪✳ ❈♦♠♦ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❙✉❜✲ ❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ r❡str✐çõ❡s ❞✐s❥✉♥t✐✈❛s é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞♦ ❙❆▼ ✭♦s ♣r♦❜❧❡♠❛s sã♦ ✐❞ê♥t✐❝♦s q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ r❡str✐çõ❡s é ✈❛③✐♦✮✱ ❙❆▼❘❉◆ ❡ ❙❆▼❘❉P t❛♠❜é♠ ♣❡rt❡♥❝❡♠ à ❝❧❛ss❡ N P✕❉✐❢í❝✐❧✳
❯♠ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐sã♦ é ✉♠❛ ♣❡r❣✉♥t❛ s♦❜r❡ ✉♠ s✐st❡♠❛ ❢♦r♠❛❧ ❡ ✉♠ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ❞❡ ❡♥tr❛❞❛s ❝✉❥❛ r❡s♣♦st❛ é ❜✐♥ár✐❛✿ s✐♠ ♦✉ ♥ã♦✳ ❯♠ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐sã♦
D s❡ tr❛♥s❢♦r♠❛ ♣♦❧✐♥♦♠✐❛❧♠❡♥t❡ ❡♠ ✉♠ ♦✉tr♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐sã♦ D′ s❡ s❡♠♣r❡ é
♣♦ssí✈❡❧✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ✐♥stâ♥❝✐❛ ❞❡ D✱ ID✱ ❝♦♥str✉✐r ✉♠❛ ✐♥stâ♥❝✐❛ ❞❡ D′✱ ID′✱ ❡♠
t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ ❡♠ r❡❧❛çã♦ ❛♦ t❛♠❛♥❤♦ ❞❛ ✐♥stâ♥❝✐❛ ❞❡D✱ t❛❧ q✉❡ID t❡♠ s♦❧✉çã♦ s✐♠
s❡ ❡ s♦♠❡♥t❡ s❡✱ID′ t❡♠ s♦❧✉çã♦ s✐♠✳ ❙❡Ds❡ tr❛♥s❢♦r♠❛ ✭♦✉ s❡ r❡❞✉③✮ ♣♦❧✐♥♦♠✐❛❧♠❡♥t❡
❛ D′ ❡♥tã♦✱ s❡ ❡①✐st❡ ✉♠ ❛❧❣♦r✐t♠♦ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ r❡s♦❧✈❡r
D′✱ t❛♠❜é♠ ❡①✐st✐rá ✉♠
❛❧❣♦r✐t♠♦ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ r❡s♦❧✈❡r D ❬❑❛r♣✱ ✶✾✼✷❪ ✳
❯♠ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐sã♦ D ∈ N P é N P✕❈♦♠♣❧❡t♦ s❡ t♦❞♦ ♣r♦❜❧❡♠❛ ❡♠ N P ♣✉❞❡r s❡r r❡❞✉③✐❞♦ ❛ D✳ ❯♠ ♣r♦❜❧❡♠❛ H é N P✕❉✐❢í❝✐❧ s❡ ❡①✐st❡ ✉♠ ♣r♦❜❧❡♠❛ N P✕ ❈♦♠♣❧❡t♦ L q✉❡ s❡ r❡❞✉③ ♣♦❧✐♥♦♠✐❛❧♠❡♥t❡ ❛H✳ ❙❡❣✉❡ ❛ ❞❡✜♥✐çã♦ ❞♦s ♣r✐♥❝✐♣❛✐s ♣r♦✲
❜❧❡♠❛s N P✕❈♦♠♣❧❡t♦s r❡❢❡r❡♥❝✐❛❞♦s ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❉❡✜♥✐çã♦ ✷✸✳ ❙❛t✐s❢❛❜✐❧✐❞❛❞❡ ♦✉ ❙❆❚ ✭❙❛t✐s✜❛❜✐❧✐t②✮✿ ❞❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ C ❞❡ ❝❧á✉✲
✽ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦
{u1, u2, . . . , un}✱ ❛ ♣❡r❣✉♥t❛ q✉❡ s❡ ❞❡s❡❥❛ r❡s♣♦♥❞❡r é✿ ❡①✐st❡ ✉♠❛ ❛tr✐❜✉✐çã♦ ❞❡ ✈❛✲
❧♦r❡s ❜✐♥ár✐♦s às ✈❛r✐á✈❡✐s ❞❡ ❢♦r♠❛ q✉❡ t♦❞❛s ❛s ❝❧á✉s✉❧❛s s❡❥❛♠ s❛t✐s❢❡✐t❛s❄
❉❡✜♥✐çã♦ ✷✹✳ ✸✲❙❛t✐s❢❛❜✐❧✐❞❛❞❡ ♦✉ ✸✲❙❆❚ ✭✸✲❙❛t✐s✜❛❜✐❧✐t②✮✿ ❞❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ C ❞❡
❝❧á✉s✉❧❛s✱ ❝❛❞❛ ✉♠❛ ❢♦r♠❛❞❛ ♣❡❧❛ ❞✐s❥✉♥çã♦ ❞❡ três ❧✐t❡r❛✐s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s
U = {u1, u2, . . . , un}✱ ❛ ♣❡r❣✉♥t❛ q✉❡ s❡ ❞❡s❡❥❛ r❡s♣♦♥❞❡r é✿ ❡①✐st❡ ✉♠❛ ❛tr✐❜✉✐çã♦ ❞❡
✈❛❧♦r❡s ❜✐♥ár✐♦s às ✈❛r✐á✈❡✐s ❞❡ ❢♦r♠❛ q✉❡ t♦❞❛s ❛s ❝❧á✉s✉❧❛s s❡❥❛♠ s❛t✐s❢❡✐t❛s❄
❖s ♣r♦❜❧❡♠❛s ❞❡ ❞❡❝✐sã♦ ❙❆❚ ❡ ✸✲❙❆❚ ❢♦r❛♠ ♣r♦✈❛❞♦s ♣❡rt❡♥❝❡♥t❡s à ❝❧❛ss❡N P✲ ❈♦♠♣❧❡t♦ ♣♦r ❈♦♦❦ ❬✶✾✼✶❪✳
❉❡✜♥✐çã♦ ✷✺✳ ❈❧✐q✉❡✿ ❞❛❞♦ ✉♠ ❣r❛❢♦ G = (V, E) ❡ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ k ≤ |V| ❝♦♠♦ ✉♠❛ ✐♥stâ♥❝✐❛✱ ❛ ♣❡r❣✉♥t❛ q✉❡ s❡ ❞❡s❡❥❛ r❡s♣♦♥❞❡r é✿ ❡①✐st❡ ✉♠❛ ❝❧✐q✉❡ ❡♠ G ❞❡
t❛♠❛♥❤♦ ♠❛✐♦r q✉❡ ♦✉ ✐❣✉❛❧ ❛ k✱ ✐st♦ é✱ ✉♠ s✉❜❝♦♥❥✉♥t♦V′ ⊆
V ❝♦♠ |V′| ≥
k✱ t❛❧ q✉❡
t♦❞❛ ❞✉♣❧❛ ❞❡ ✈ért✐❝❡s ❡♠ V′ ❡stá ❧✐❣❛❞❛ ♣♦r ✉♠❛ ❛r❡st❛ ❡♠
E❄
❉❡✜♥✐çã♦ ✷✻✳ ❈♦❜❡rt✉r❛ ♣♦r ❱ért✐❝❡s ✭❈❱✮✿ ❞❛❞♦ ✉♠ ❣r❛❢♦ G= (V, E) ❡ ✉♠ ✐♥t❡✐r♦
♣♦s✐t✐✈♦k ≤ |V|❝♦♠♦ ✉♠❛ ✐♥stâ♥❝✐❛✱ ❛ ♣❡r❣✉♥t❛ q✉❡ s❡ ❞❡s❡❥❛ r❡s♣♦♥❞❡r é✿ ❡①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ ♣♦r ✈ért✐❝❡s ❡♠ G ❞❡ t❛♠❛♥❤♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ k✱ ✐st♦ é✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ V′ ⊆
V t❛❧ q✉❡ |V′| ≤
k ❡✱ ♣❛r❛ ❝❛❞❛ ❛r❡st❛ {u, v} ∈E✱ u∈V′ ♦✉
v ∈V′❄
❉❡✜♥✐çã♦ ✷✼✳ ❈♦♥❥✉♥t♦ ■♥❞❡♣❡♥❞❡♥t❡ ✭❈■✮✿ ❞❛❞♦ ✉♠ ❣r❛❢♦ G = (V, E) ❡ ✉♠ ✐♥t❡✐r♦
♣♦s✐t✐✈♦ k≤ |V| ❝♦♠♦ ✉♠❛ ✐♥stâ♥❝✐❛✱ ❛ ♣❡r❣✉♥t❛ q✉❡ s❡ ❞❡s❡❥❛ r❡s♣♦♥❞❡r é✿ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❡♠G❞❡ t❛♠❛♥❤♦ ♠❛✐♦r q✉❡ ♦✉ ✐❣✉❛❧ ❛k✱ ✐st♦ é✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ V′ ⊆
V t❛❧ q✉❡ |V′| ≥
k ❡✱ ♣❛r❛ t♦❞♦ u, v ∈V′✱ ❛ ❛r❡st❛ {
u, v}∈/ E❄
❑❛r♣ ❬✶✾✼✷❪ ♣r♦✈♦✉✱ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ✭❛ ♣❛rt✐r ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❙❛t✐s❢❛❜✐❧✐❞❛❞❡✱ tr❛♥s❢♦r♠❛❞♦ ❡♠ ❈❧✐q✉❡ q✉❡ é ❡♥tã♦ r❡❞✉③✐❞♦ ❛ ❈♦❜❡r✲ t✉r❛ ♣♦r ❱ért✐❝❡s✮✱ q✉❡ ❛ ✈❡rsã♦ ❞❡ ❞❡❝✐sã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈♦❜❡rt✉r❛ ♣♦r ❱ért✐❝❡s é N P✲❈♦♠♣❧❡t♦✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣r♦✈♦✉ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐sã♦ ❞❡ ❈♦♥❥✉♥t♦ ■♥❞❡♣❡♥❞❡♥t❡ é t❛♠❜é♠ N P✕❈♦♠♣❧❡t♦✱ ✉♠❛ ✈❡③ ❝♦♥❤❡❝✐❞❛ ❛ r❡❧❛çã♦✿ V′ é ✉♠❛ ❝♦✲
❜❡rt✉r❛ ♣♦r ✈ért✐❝❡s ❡♠ G✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ V −V′ é ✉♠ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡
G✳
❉❡✜♥✐çã♦ ✷✽✳ ❙✉❜❣r❛❢♦ ❉✐r❡❝✐♦♥❛❞♦ ❆❝í❝❧✐❝♦✿ ❞❛❞♦ ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦G= (V, A)❡
✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ k≤ |A| ❝♦♠♦ ✉♠❛ ✐♥stâ♥❝✐❛✱ ❛ ♣❡r❣✉♥t❛ q✉❡ s❡ ❞❡s❡❥❛ r❡s♣♦♥❞❡r é✿ ❡①✐st❡ ✉♠ s✉❜❣r❛❢♦ ❛❝í❝❧✐❝♦ ❡♠G❞❡ t❛♠❛♥❤♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛k✱ ✐st♦ é✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ A′ ⊆
A t❛❧ q✉❡ |A′| ≥
k✱ ❡ q✉❡ ♦ s✉❜❣r❛❢♦ ✐♥❞✉③✐❞♦ G[A′
] s❡❥❛ ❛❝í❝❧✐❝♦❄
❉❡✜♥✐çã♦ ✷✾✳ ❈♦♥❥✉♥t♦ ❞❡ ❆r❝♦s ❞❡ ❘❡t♦r♥♦✿ ❞❛❞♦ ✉♠ ❣r❛❢♦ ❞✐r❡❝✐♦♥❛❞♦G= (V, A)
✶✳✸✳ ❆❜♦r❞❛❣❡♥s ♣❛r❛ ♣r♦❜❧❡♠❛s N P✕❉✐❢í❝❡✐s ✾
é✿ ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A′ ⊆ A t❛❧ q✉❡ |A′| ≤ k✱ ❡ q✉❡ A′ ❝♦♥t❡♥❤❛ ❛♦ ♠❡♥♦s ✉♠
❛r❝♦ ❞❡ ❝❛❞❛ ❝✐❝❧♦ ❡♠ G❄
❑❛r♣ ❬✶✾✼✷❪ ♣r♦✈♦✉ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈♦♥❥✉♥t♦ ❞❡ ❆r❝♦s ❞❡ ❘❡t♦r♥♦ ✭❞♦ ✐♥❣❧ês ❋❡❡❞❜❛❝❦ ❆r❝ ❙❡t Pr♦❜❧❡♠✮ é N P✲❈♦♠♣❧❡t♦ ❢❛③❡♥❞♦ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❛ ♣❛rt✐r ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈♦❜❡rt✉r❛ ♣♦r ❱ért✐❝❡s✳ ❚❡♠✲s❡ ❛ r❡❧❛çã♦✿ A′ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛r❝♦s
❞❡ r❡t♦r♥♦ ❞❡ G✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ G[A\A′
] é ✉♠ s✉❜❣r❛❢♦ ❛❝í❝❧✐❝♦✳ ❆ss✐♠ s❡♥❞♦✱ ♦
♣r♦❜❧❡♠❛ ❞♦ ❙✉❜❣r❛❢♦ ❉✐r❡❝✐♦♥❛❞♦ ❆❝í❝❧✐❝♦ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣r♦❜❧❡♠❛ ❞♦ ❈♦♥❥✉♥t♦ ❞❡ ❆r❝♦s ❞❡ ❘❡t♦r♥♦✳
▲♦❣♦✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♥❝♦♥tr❛r ✉♠ ❙✉❜❣r❛❢♦ ❉✐r❡❝✐♦♥❛❞♦ ❆❝í❝❧✐❝♦ t❛♠❜é♠ é ♣r♦✲ ✈❛❞♦N P✕❈♦♠♣❧❡t♦✳ ❖ ♣r♦❜❧❡♠❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ♣❛r❛ ❣r❛❢♦s ♥ã♦ ❞✐r❡❝✐♦♥❛❞♦s é r❡s♦❧✲ ✈✐❞♦ tr✐✈✐❛❧♠❡♥t❡ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ ❬❑❛r♣✱ ✶✾✼✷❪✳
✶✳✸ ❆❜♦r❞❛❣❡♥s ♣❛r❛ ♣r♦❜❧❡♠❛s
N P
✕❉✐❢í❝❡✐s
❊①✐st❡♠ ♣❡❧♦ ♠❡♥♦s três ❛❜♦r❞❛❣❡♥s ♣❛r❛ tr❛t❛r ♣r♦❜❧❡♠❛s ❞❛ ❝❧❛ss❡ N P✕❉✐❢í❝✐❧✳ Pr✐✲ ♠❡✐r❛♠❡♥t❡✱ s❡ ❛s ✐♥stâ♥❝✐❛s sã♦ ♣❡q✉❡♥❛s✱ ❛❧❣♦r✐t♠♦s ❡①❛t♦s ❡①♣♦♥❡♥❝✐❛✐s✱ q✉❡ ❜✉s❝❛♠ r❡s♦❧✈❡r ♥❛ ♦t✐♠❛❧✐❞❛❞❡ ♦s ♣r♦❜❧❡♠❛s✱ ♣♦❞❡♠ s❡r s❛t✐s❢❛tór✐♦s✳
❖✉tr❛ ❛❜♦r❞❛❣❡♠ é ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ♣ró①✐♠❛s ❛♦ ót✐♠♦✱ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧✱ ♣♦r ♠❡✐♦ ❞❡ té❝♥✐❝❛s ❤❡✉ríst✐❝❛s ♦✉ ❛❧❣♦r✐t♠♦s ❛♣r♦①✐♠❛t✐✈♦s✳ ❆❧❣♦r✐t♠♦s ❛♣r♦①✐♠❛✲ t✐✈♦s sã♦ ❛q✉❡❧❡s q✉❡ r❡t♦r♥❛♠ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦ ❝✉❥♦ ✈❛❧♦r ❡stá ❛ ✉♠ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♥❤❡❝✐❞♦ ❞❛ s♦❧✉çã♦ ót✐♠❛✳ ❙❡ ❡st❡ ❢❛t♦r é ❝♦♥st❛♥t❡✱ ♦✉ s❡❥❛✱ ✐♥❞❡♣❡♥❞❡ ❞♦ t❛♠❛♥❤♦ ❞❛ ✐♥stâ♥❝✐❛✱ t❡♠✲s❡ ✉♠ ❛❧❣♦r✐t♠♦ ❛♣r♦①✐♠❛t✐✈♦ q✉❡ ♣r♦❞✉③ ✉♠❛ s♦❧✉çã♦ ❝✉❥♦ ✈❛❧♦r ❡stá ❛ ✉♠ ❝❡rt♦ ❢❛t♦r ❝♦♥st❛♥t❡ ❞❛ s♦❧✉çã♦ ót✐♠❛✳
❍❡✉ríst✐❝❛s sã♦ té❝♥✐❝❛s ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ✈✐s❛♠ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧✱ s❡ ♣♦ssí✈❡❧ ❞❡ ❜♦❛ q✉❛❧✐❞❛❞❡✱ ❡♠ ✉♠ t❡♠♣♦ ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ r❛③♦á✈❡❧✱ ♥ã♦ ❣❛r❛♥t✐♥❞♦ ❛ s✉❛ ♦t✐♠❛❧✐❞❛❞❡ ❡ ♥❡♠ s❡q✉❡r ✉♠ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦✳
✶✳✹ ❖❜❥❡t✐✈♦s ❞♦ tr❛❜❛❧❤♦
❉❡s❡❥❛✲s❡ ✐♥✈❡st✐❣❛r ♥❡st❡ tr❛❜❛❧❤♦ r❡s✉❧t❛❞♦s t❡ór✐❝♦s ❡ ❡①♣❡r✐♠❡♥t❛✐s r❡❧❛❝✐♦♥❛❞♦s ❛♦s ♣r♦❜❧❡♠❛s ❞♦ ❙✉❜❣r❛❢♦ ❆❝í❝❧✐❝♦ ▼á①✐♠♦ s♦❜ ❘❡str✐çõ❡s ❉✐s❥✉♥t✐✈❛s P♦s✐t✐✈❛s ❡ ◆❡❣❛t✐✈❛s✳ ❙❡♥❞♦ ❛ss✐♠✱ ♦s s❡❣✉✐♥t❡s ♦❜❥❡t✐✈♦s sã♦ ♣r♦♣♦st♦s✿
• ■♥✈❡st✐❣❛r ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞♦s ♣r♦❜❧❡♠❛s❀
