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▼❛t❤✷✸✻ ❉✐s❝r❡t❡ ▼❛t❤s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s

P✳ ■tt♠❛♥♥

❯❑❩◆✱ P✐❡t❡r♠❛r✐t③❜✉r❣

❙❡♠❡st❡r ✶✱ ✷✵✶✷

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶ ✴ ✷✹

❍❛✈❡❧✲❍❛❦✐♠✐

❚❤❡♦r❡♠ ▲❡t ❉ ❜❡ t❤❡ s❡q✉❡♥❝❡ ❞✶, ❞✷, . . . , ❞♥ ✇✐t❤ ❞✶ ≥ ❞✷ ≥ · · · ≥ ❞♥ ❛♥❞ ♥ ≥ ✷ ▲❡t ❉′ ❜❡ t❤❡ s❡q✉❡♥❝❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❉ ❜②

✶

❉✐s❝❛r❞✐♥❣ ❞✶✱ ❛♥❞✱

✷

❙✉❜tr❛❝t✐♥❣ ✶ ❢r♦♠ ❡❛❝❤ ♦❢ t❤❡ ♥❡①t ❞✶ ❡♥tr✐❡s ♦❢ ❉

❚❤❛t ✐s✱ ❉′ ✐s t❤❡ s❡q✉❡♥❝❡ ❞✷ − ✶, ❞✸ − ✶, . . . , ❞❞✶+✶ − ✶, ❞❞✶+✷, . . . , ❞♥ ❚❤❡♥✱ ❉ ✐s ❣r❛♣❤✐❝❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❉′ ✐s ❣r❛♣❤✐❝❛❧

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷ ✴ ✷✹

❍❛✈❡❧✲❍❛❦✐♠✐ ✭❝♦♥t✳✮

Pr♦♦❢✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ s❡q✉❡♥❝❡ ❉′ ✐s ❣r❛♣❤✐❝❛❧ ▲❡t ●✶ ❜❡ ❛ ❣r❛♣❤ ♦❢ ♦r❞❡r ♥ − ✶ ✇✐t❤ ❞❡❣r❡❡ s❡q✉❡♥❝❡ ❉′ ❚❤❡♥ t❤❡ ✈❡rt✐❝❡s ♦❢ ●✶ ❝❛♥ ❜❡ ❧❛❜❡❧❧❡❞ ❛s ✈✷, ✈✸, . . . , ✈♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ❞❡❣● ′(✈✐) = ❞✐ − ✶ ✐❢ ✷ ≤ ✐ ≤ ❞✶ + ✶ ❛♥❞ ❞(✈✐) = ❞✐ ✐❢ ❞✶ + ✷ ≤ ✐ ≤ ♥ ❲❡ ❝❛♥ ♥♦✇ ❝♦♥str✉❝t ❛ ♥❡✇ ❣r❛♣❤ ● ❢r♦♠ ●✶ ❜② ❛❞❞✐♥❣ ❛ ♥❡✇ ✈❡rt❡① ✈✶ ❛♥❞ t❤❡♥ ❥♦✐♥✐♥❣ ✈✶ ✇✐t❤ ❛♥ ❡❞❣❡ t♦ ❡❛❝❤ ♦❢ ✈✷, ✈✸, . . . , ✈❞✶+✶

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✸ ✴ ✷✹

❍❛✈❡❧✲❍❛❦✐♠✐ ✭❝♦♥t✳✮

Pr♦♦❢✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ✈✶ ✐s ❞✶ ❚❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ♦t❤❡r ✈❡rt✐❝❡s ❛r❡ t❤❡ r❡♠❛✐♥✐♥❣ ✈❛❧✉❡s ♦❢ ❉ ❚❤✉s✱ ✇❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ ❛ ❣r❛♣❤ ✇✐t❤ ❞❡❣r❡❡ s❡q✉❡♥❝❡ ❉✱ ❛♥❞ s♦ ❉ ✐s ❣r❛♣❤✐❝❛❧ ❲❡ s❤♦✇ ♥❡①t t❤❛t ✐❢ ❉ ✐s ❣r❛♣❤✐❝❛❧✱ t❤❡♥ ❉′ ✐s ❣r❛♣❤✐❝❛❧ ❆ss✉♠❡ t❤❛t ❉ ✐s ❛ ❣r❛♣❤✐❝❛❧ s❡q✉❡♥❝❡ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ❛r❡ ♦♥❡ ♦r ♠♦r❡ ❣r❛♣❤s ♦❢ ♦r❞❡r ♥ ✇✐t❤ ❞❡❣r❡❡ s❡q✉❡♥❝❡ ❉

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✹ ✴ ✷✹

❍❛✈❡❧✲❍❛❦✐♠✐ ✭❝♦♥t✳✮

Pr♦♦❢✳ ❆♠♦♥❣ ❛❧❧ s✉❝❤ ❣r❛♣❤s✱ ❧❡t ● ❜❡ ♦♥❡ s✉❝❤ t❤❛t ❱ (●) = {✈✶, ✈✷, . . . , ✈♥}✱ ✇❤❡r❡ ❞❡❣●(✈✐) = ❞✐ ❢♦r ✶ ≤ ✐ ≤ ♥✱ ❛♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ✈❡rt✐❝❡s ❛❞❥❛❝❡♥t ✇✐t❤ ✈✶ ✐s ♠❛①✐♠✉♠ ❲❡ s❤♦✇ t❤❛t ✐♥ ●✱ t❤❡ ✈❡rt❡① ✈✶ ♠✉st ❜❡ ❛❞❥❛❝❡♥t ✇✐t❤ ✈❡rt✐❝❡s ❤❛✈✐♥❣ ❞❡❣r❡❡s ❞✷, ❞✸, . . . , ❞❞✶+✶ ❙✉♣♣♦s❡ t❤✐s ✐s ♥♦t t❤❡ ❝❛s❡ ❚❤❡r❡ ♠✉st ❡①✐st t✇♦ ✈❡rt✐❝❡s ✈❥ ❛♥❞ ✈❦ ✇✐t❤ ❞❥ > ❞❦ s✉❝❤ t❤❛t ✈✶ ✐s ❛❞❥❛❝❡♥t t♦ ✈❦✱ ❜✉t ♥♦t t♦ ✈❥ ❙✐♥❝❡ t❤❡ ❞❡❣r❡❡ ♦❢ ✈❥ ❡①❝❡❡❞s t❤❛t ♦❢ ✈❦✱ t❤❡r❡ ♠✉st ❜❡ s♦♠❡ ✈❡rt❡① ✈ℓ s✉❝❤ t❤❛t ✈ℓ ✐s ❛❞❥❛❝❡♥t t♦ ✈❥✱ ❜✉t ♥♦t t♦ ✈❦

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✺ ✴ ✷✹

❍❛✈❡❧✲❍❛❦✐♠✐ ✭❝♦♥t✳✮

Pr♦♦❢✳ ❘❡♠♦✈✐♥❣ t❤❡ ❡❞❣❡s ✈✶✈❦ ❛♥❞ ✈❥✈ℓ ❛♥❞ ❛❞❞✐♥❣ t❤❡ ❡❞❣❡s ✈✶✈❥ ❛♥❞ ✈❦✈ℓ ♣r♦❞✉❝❡s ❛ ♥❡✇ ❣r❛♣❤ ❍ t❤❛t ❛❧s♦ ❤❛s ❞❡❣r❡❡ s❡q✉❡♥❝❡ ❉ ❍♦✇❡✈❡r✱ ✐♥ ❍ t❤❡ s✉♠ ♦❢ t❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ✈❡rt✐❝❡s ❛❞❥❛❝❡♥t ✇✐t❤ ✈✶ ✐s ❧❛r❣❡r t❤❛♥ t❤❛t ✐♥ ●✱ ❝♦♥tr❛❞✐❝t✐♥❣ ♦✉r ❝❤♦✐❝❡ ♦❢ ● ❚❤✐s ❝♦♥tr❛❞✐❝t✐♦♥ ❛r❣✉♠❡♥t ✈❡r✐✜❡s t❤❛t ♦✉r ✐♥✐t✐❛❧ ❛ss✉♠♣t✐♦♥ t❤❛t ✈✶ ✐s ♥♦t ❛❞❥❛❝❡♥t ✇✐t❤ ✈❡rt✐❝❡s ❤❛✈✐♥❣ ❞❡❣r❡❡s ❞✷, ❞✸, . . . , ❞❞✶+✶✮ ✇❛s ❢❛❧s❡ ❚❤✉s✱ ❛s ❝❧❛✐♠❡❞✱ ✈✶ ♠✉st ❜❡ ❛❞❥❛❝❡♥t ✇✐t❤ ✈❡rt✐❝❡s ❤❛✈✐♥❣ ❞❡❣r❡❡s ❞✷, ❞✸, . . . , ❞❞✶+✶ ❍❡♥❝❡✱ t❤❡ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❢r♦♠ ● ❜② r❡♠♦✈✐♥❣ ✈✶✱ t♦❣❡t❤❡r ✇✐t❤ ❛❧❧ t❤❡ ❡❞❣❡s ✐♥❝✐❞❡♥t ✇✐t❤ ✈✶✱ ♣r♦❞✉❝❡s ❛ ❣r❛♣❤ ✇✐t❤ ❞❡❣r❡❡ s❡q✉❡♥❝❡ ❉′✱ s♦ ❉′ ✐s ❣r❛♣❤✐❝❛❧

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✻ ✴ ✷✹

❙✉❜❣r❛♣❤s

❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤s

G: u v w x y H: u v w x I:

❍ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ●✱ ❜✉t ■ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ●

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✼ ✴ ✷✹

❙✉❜❣r❛♣❤s ✭❝♦♥t✳✮

❉❡✜♥✐t✐♦♥ ❆ s✉❜❣r❛♣❤ ♦❢ ❛ ❣r❛♣❤ ● ✐s ❛ ❣r❛♣❤ ❛❧❧ ♦❢ ✇❤♦s❡ ✈❡rt✐❝❡s ❜❡❧♦♥❣ t♦ ❱ (●) ❛♥❞ ❛❧❧ ♦❢ ✇❤♦s❡ ❡❞❣❡s ❜❡❧♦♥❣ t♦ ❊(●) ■❢ ❍ ✐s ❛ s✉❜❣r❛♣❤ ♦❢ ●✱ t❤❡♥ ✇❡ ✇r✐t❡ ❍ ⊂ ● ■❢ ❛ s✉❜❣r❛♣❤ ❍ ♦❢ ● ❝♦♥t❛✐♥s ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ ●✱ t❤❡♥ ❍ ✐s ❝❛❧❧❡❞ ❛ s♣❛♥♥✐♥❣ s✉❜❣r❛♣❤ ♦❢ ●

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✽ ✴ ✷✹

❙✉❜❣r❛♣❤s ✭❝♦♥t✳✮

❊①❛♠♣❧❡ ■♥ t❤❡ ♣r❡✈✐♦✉s ❞✐❛❣r❛♠✱ ❍ ✐s ❛ s✉❜❣r❛♣❤ ♦❢ ●✱ ❜✉t ♥♦t ❛ s♣❛♥♥✐♥❣ s✉❜❣r❛♣❤

G: u v w x y H: u v w x J: u v x y w

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❏ ✐s ❛ s♣❛♥♥✐♥❣ s✉❜❣r❛♣❤ ♦❢ ● s✐♥❝❡ ❱ (●) = ❱ (❏)

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✾ ✴ ✷✹

❙✉❜❣r❛♣❤s ✭❝♦♥t✳✮

■❢ ❲ ✐s ❛ ♥♦♥❡♠♣t② s✉❜s❡t ♦❢ ✈❡rt✐❝❡s ♦❢ ❛ ❣r❛♣❤ ●✱ t❤❡♥ t❤❡ s✉❜❣r❛♣❤

• [❲ ] ♦❢ ● ✐♥❞✉❝❡❞ ❜② ❲ ✐s t❤❡ ❣r❛♣❤ ❤❛✈✐♥❣ ✈❡rt❡① s❡t ❲ ❛♥❞

✇❤♦s❡ ❡❞❣❡ s❡t ❝♦♥s✐sts ♦❢ ❛❧❧ t❤♦s❡ ❡❞❣❡s ♦❢ ● ✐♥❝✐❞❡♥t ✇✐t❤ t✇♦ ✈❡rt✐❝❡s ✐♥ ❲ ❆ s✉❜❣r❛♣❤ ❍ ♦❢ ❛ ● ✐s ❝❛❧❧❡❞ ❛♥ ✈❡rt❡①✲✐♥❞✉❝❡❞ s✉❜❣r❛♣❤✱ ♦r s✐♠♣❧② ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤✱ ♦❢ ● ✐❢ ❍ = ●[❲ ] ❢♦r s♦♠❡ s✉❜s❡t ❲ ♦❢ ❱ (●) ❍❡♥❝❡ ✐❢ ❍ ✐s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ ●✱ t❤❡♥ ❡✈❡r② ❡❞❣❡ ♦❢ ● ✐♥❝✐❞❡♥t ✇✐t❤ t✇♦ ✈❡rt✐❝❡s ✐♥ ❱ (❍) ❜❡❧♦♥❣s t♦ ❊(❍) ❚❤✐s ♠❡❛♥s t❤❛t t✇♦ ✈❡rt✐❝❡s ❛r❡ ❛❞❥❛❝❡♥t ✐♥ ❍ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡② ❛r❡ ❛❞❥❛❝❡♥t ✐♥ ●

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✵ ✴ ✷✹

❙✉❜❣r❛♣❤s ✭❝♦♥t✳✮

❙✐♠✐❧❛r❧②✱ ✐❢ ❋ ✐s ❛ ♥♦♥❡♠♣t② s✉❜s❡t ♦❢ ❡❞❣❡s ♦❢ ●✱ t❤❡♥ t❤❡ s✉❜❣r❛♣❤

• [❋] ✐♥❞✉❝❡❞ ❜② ❋ ✐s t❤❡ ❣r❛♣❤ ✇❤♦s❡ ✈❡rt❡① s❡t ❝♦♥s✐sts ♦❢ ❛❧❧ t❤♦s❡

✈❡rt✐❝❡s ♦❢ ● ✐♥❝✐❞❡♥t ✇✐t❤ ❛♥ ❡❞❣❡ ✐♥ ❋ ❛♥❞ ✇❤♦s❡ ❡❞❣❡ s❡t ✐s ❋ ❆ s✉❜❣r❛♣❤ ❍ ♦❢ ❛ ● ✐s ❝❛❧❧❡❞ ❛♥ ❡❞❣❡✲✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ ● ✐❢ ❍ = ●[❋] ❢♦r s♦♠❡ s✉❜s❡t ❋ ♦❢ ❊(●)

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✶ ✴ ✷✹

❙✉❜❣r❛♣❤s ✭❝♦♥t✳✮

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤s

G: u v w x y H: u v w x I: v x y w

◆♦t❡ t❤❛t ❍ ✐s ♥♦t ❛ ✈❡rt❡①✲✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ ●✱ ❜✉t ✐s ✐s ❛♥ ❡❞❣❡✲✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ■ ✐s ❛ ✈❡rt❡① ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ ●✱ ✐♥ ❢❛❝t ■ = ●[{✈, ✇, ①, ②}] ■s ■ ❛♥ ❡❞❣❡✲✐♥❞✉❝❡❞ s✉❜❣r❛♣❤❄

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✷ ✴ ✷✹

❙♦♠❡ ❣r❛♣❤ ♦♣❡r❛t✐♦♥s

❉❡✜♥✐t✐♦♥ ▲❡t ● ❜❡ ❛ ❣r❛♣❤ ❛♥❞ ❧❡t ❯ ⊆ ❊(●) ❚❤❡♥ ● − ❯ ❞❡♥♦t❡s t❤❡ s✉❜❣r❛♣❤ ♦❢ ● ♦❜t❛✐♥❡❞ ❢r♦♠ ● ❜❡ ❞❡❧❡t✐♥❣ t❤❡ ❡❞❣❡s ♦❢ ❯ ▲❡t ❙ ⊂ ❱ (●) ❚❤❡♥ ● − ❙ ❞❡♥♦t❡s s✉❜❣r❛♣❤ ♦❢ ● ♦❜t❛✐♥❡❞ ❢r♦♠ ● ❜② ❞❡❧❡t✐♥❣ t❤❡ ✈❡rt✐❝❡s ♦❢ ❙ ❛♥❞ ❛❧❧ ❡❞❣❡s ✐♥❝✐❞❡♥t ✇✐t❤ ❛ ✈❡rt❡① ✐♥ ❙

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✸ ✴ ✷✹

❙♦♠❡ ❣r❛♣❤ ♦♣❡r❛t✐♦♥s ✭❝♦♥t✳✮

❈♦♥s✐❞❡r ❣r❛♣❤ ●

G: a b c d x y

❋✐♥❞

✶

• − {❞, ❛}

✷

• − {❜❞, ❛❜}

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✹ ✴ ✷✹

❲❛❧❦s

• r❛♣❤s ♦❢t❡♥ r❡♣r❡s❡♥t ♣❤②s✐❝❛❧ ♥❡t✇♦r❦s ✭r♦❛❞s✱ ▲❆◆s✱ ❡t❝✳✮

❍♦✇ ❞♦ ✇❡ ❞❡s❝r✐❜❡ ♠♦✈❡♠❡♥t t❤r♦✉❣❤ ❛ ❣r❛♣❤❄ ❉❡✜♥✐t✐♦♥ ❆ ✇❛❧❦ ✐♥ ❛ ❣r❛♣❤ ● ✐s ❛ s❡q✉❡♥❝❡ ❲ : ✈✶, ✈✷, . . . , ✈❦ s✉❝❤ t❤❛t ✈✐✈✐+✶ ∈ ❊(●) ❢♦r ✐ = ✶, ✷, . . . , ❦ − ✶ ❲❡ ❝❛❧❧ ❲ ❛ ✈✶ − ✈❦ ✇❛❧❦ ❚❤❡ ❧❡♥❣t❤ ♦❢ ❲ ✐s ❦ − ✶

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✺ ✴ ✷✹

❲❛❧❦s ✭❝♦♥t✳✮

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤ ●

G: a b c d x y

❲ : ❜, ❞, ❝, ❞, ①, ② ✐s ❛ ✇❛❧❦ ♦❢ ❧❡♥❣t❤ ✺ ❲❡ ❛❜❜r❡✈✐❛t❡ t❤✐s ✇❛❧❦ ❛s ❜❞❝❞①②

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✻ ✴ ✷✹

❈♦♥♥❡❝t❡❞♥❡ss

❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤s ● ❛♥❞ ❍

G: H: x v y

◆♦t❡ t❤❛t ● ❤❛s ✏♦♥❡ ♣✐❡❝❡✑✱ ✇❤✐❧❡ ❍ ❤❛s ✏t❤r❡❡ ♣✐❡❝❡s✑ ❲❡ ❢♦r♠❛❧✐s❡ t❤✐s ✐❞❡❛ ❛s ❢♦❧❧♦✇s

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✼ ✴ ✷✹

❈♦♥♥❡❝t❡❞♥❡ss ✭❝♦♥t✳✮

❉❡✜♥✐t✐♦♥ ▲❡t ✉, ✈ ❜❡ ✈❡rt✐❝❡s ♦❢ ❛ ❣r❛♣❤ ● ■❢ t❤❡r❡ ❡①✐sts ❛ ✉ − ✈ ♣❛t❤ ✐♥ ●✱ ✇❡ s❛② t❤❛t ✉ ❛♥❞ ✈ ❛r❡ ❝♦♥♥❡❝t❡❞ ■♥ t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡✱ ① ❛♥❞ ② ❛r❡ ❝♦♥♥❡❝t❡❞✱ ❜✉t ① ❛♥❞ ✈ ❛r❡ ♥♦t ❝♦♥♥❡❝t❡❞ ❲❡ ❝❛❧❧ ● ❛ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✐❢ ❡✈❡r② t✇♦ ✈❡rt✐❝❡s ♦❢ ● ❛r❡ ❝♦♥♥❡❝t❡❞ ❆ ❝♦♠♣♦♥❡♥t ♦❢ ❛ ❣r❛♣❤ ✐s ❛ ♠❛①✐♠❛❧ s✉❜❣r❛♣❤ t❤❛t ✐s ❝♦♥♥❡❝t❡❞ ✭✐✳❡✳✱ ❛ ✏♣✐❡❝❡✑✮

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✽ ✴ ✷✹

❈♦♥♥❡❝t❡❞♥❡ss ✭❝♦♥t✳✮

❊①❛♠♣❧❡

G:

■♥ t❤✐s ❝❛s❡✱ ● ✐s ♥♦t ❝♦♥♥❡❝t❡❞ ❛♥❞ ❤❛s ✸ ❝♦♠♣♦♥❡♥ts

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✶✾ ✴ ✷✹

❆❞❥❛❝❡♥❝② ♠❛tr✐❝❡s

❆♥♦t❤❡r q✉❡st✐♦♥ ✇❤✐❝❤ ❛r✐s❡s ✐♥ ❣r❛♣❤ t❤❡♦r② ✐s ❤♦✇ ❜❡st t♦ r❡♣r❡s❡♥t ❛ ❣r❛♣❤ ✐♥ ❛ ❝♦♠♣✉t❡r ❖♥❡ ✇❛② ♦❢ ❞♦✐♥❣ t❤✐s ✐s t♦ ✉s❡ ❛♥ ❛❞❥❛❝❡♥❝② ♠❛tr✐①

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✵ ✴ ✷✹

❆❞❥❛❝❡♥❝② ♠❛tr✐❝❡s ✭❝♦♥t✳✮

❉❡✜♥✐t✐♦♥ ▲❡t ● ❜❡ ❛ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t {✈✶, ✈✷, . . . , ✈♥} ❚❤❡ ❛❞❥❛❝❡♥❝② ♠❛tr✐① ♦❢ ● ✐s t❤❡ ♥ × ♥ ♠❛tr✐① ❆ = (❛✐ ❥) ✇❤❡r❡ ❛✐ ❥ =

• ✶

✐❢ ✈✐ ❛♥❞ ✈❥ ❛r❡ ❛❞❥❛❝❡♥t ✵ ✐❢ ✈✐ ❛♥❞ ✈❥ ❛r❡ ♥♦t ❛❞❥❛❝❡♥t

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✶ ✴ ✷✹

❆❞❥❛❝❡♥❝② ♠❛tr✐❝❡s ✭❝♦♥t✳✮

❊①❛♠♣❧❡ ❲❡ ❝♦♥str✉❝t t❤❡ ❛❞❥❛❝❡♥❝② ♠❛tr✐① ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤

G: v1 v2 v3 v4

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✷ ✴ ✷✹

❆❞❥❛❝❡♥❝② ♠❛tr✐❝❡s ✭❝♦♥t✳✮

❊①❛♠♣❧❡ ✈✶ ✈✷ ✈✸ ✈✹ ❆ = ✈✶ ✈✷ ✈✸ ✈✹     ✵ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✵ ✵ ✶ ✵     ❍❡♥❝❡✱ ❆ =     ✵ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✵ ✵ ✶ ✵    

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✸ ✴ ✷✹

❆❞❥❛❝❡♥❝② ♠❛tr✐❝❡s ✭❝♦♥t✳✮

❆❞❥❛❝❡♥❝② ♠❛tr✐❝❡s ❛r❡ ✉s❡❢✉❧ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❛s♦♥s

❊❛s② t♦ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦♥ ❝♦♠♣✉t❡r ▲✐♥❡❛r ❛❧❣❡❜r❛ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❣r❛♣❤s

❲❡ ♥♦t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ❛♥ ❛❞❥❛❝❡♥❝② ♠❛tr✐①

✶

❆ ✐s ❛ ✵ − ✶ ♠❛tr✐①

✷

❆ ✐s s②♠♠❡tr✐❝ ✭t❤❛t ✐s✱ ❛✐ ❥ = ❛❥ ✐ ❢♦r ❛❧❧ ✐, ❥ ∈ {✶, ✷, . . . , ♥}✮

✸

❚❤❡ ❞✐❛❣♦♥❛❧ ❡♥tr✐❡s ❛r❡ ❛❧✇❛②s ✵ ✭t❤❛t ✐s✱ ❛✐ ✐ = ✵ ❢♦r ✐✮

■tt♠❛♥♥ ✭❯❑❩◆ P▼❇✮ ▼❛t❤✷✸✻ ✷✵✶✷ ✷✹ ✴ ✷✹