rs s r rr - PowerPoint PPT Presentation

●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ▼❛♣♣✐♥❣ ❤ : ❱ ( ● ) → ❱ ( ❍ ) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝② ✿ ①② ∈ ❊ ( ● ) = ⇒ ❤ ( ① ) ❤ ( ② ) ∈ ❊ ( ❍ ) ◆♦t❛t✐♦♥✿ ● → ❍ ✳ ❚❛r❣❡t ❣r❛♣❤✿ ❍ = ❈ ✺ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✺ ✴ ✸✽

●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ▼❛♣♣✐♥❣ ❤ : ❱ ( ● ) → ❱ ( ❍ ) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝② ✿ ①② ∈ ❊ ( ● ) = ⇒ ❤ ( ① ) ❤ ( ② ) ∈ ❊ ( ❍ ) ◆♦t❛t✐♦♥✿ ● → ❍ ✳ ❚❛r❣❡t ❣r❛♣❤✿ ❍ = ❈ ✺ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✺ ✴ ✸✽

●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ▼❛♣♣✐♥❣ ❤ : ❱ ( ● ) → ❱ ( ❍ ) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝② ✿ ①② ∈ ❊ ( ● ) = ⇒ ❤ ( ① ) ❤ ( ② ) ∈ ❊ ( ❍ ) ◆♦t❛t✐♦♥✿ ● → ❍ ✳ ❚❛r❣❡t ❣r❛♣❤✿ ❍ = ❈ ✺ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✺ ✴ ✸✽

●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ▼❛♣♣✐♥❣ ❤ : ❱ ( ● ) → ❱ ( ❍ ) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝② ✿ ①② ∈ ❊ ( ● ) = ⇒ ❤ ( ① ) ❤ ( ② ) ∈ ❊ ( ❍ ) ◆♦t❛t✐♦♥✿ ● → ❍ ✳ ❚❛r❣❡t ❣r❛♣❤✿ ❍ = ❈ ✺ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✺ ✴ ✸✽

●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ▼❛♣♣✐♥❣ ❤ : ❱ ( ● ) → ❱ ( ❍ ) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝② ✿ ①② ∈ ❊ ( ● ) = ⇒ ❤ ( ① ) ❤ ( ② ) ∈ ❊ ( ❍ ) ◆♦t❛t✐♦♥✿ ● → ❍ ✳ ❚❛r❣❡t ❣r❛♣❤✿ ❍ = ❈ ✺ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✺ ✴ ✸✽

●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ▼❛♣♣✐♥❣ ❤ : ❱ ( ● ) → ❱ ( ❍ ) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝② ✿ ①② ∈ ❊ ( ● ) = ⇒ ❤ ( ① ) ❤ ( ② ) ∈ ❊ ( ❍ ) ◆♦t❛t✐♦♥✿ ● → ❍ ✳ ❚❛r❣❡t ❣r❛♣❤✿ ❍ = ❈ ✺ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✺ ✴ ✸✽

●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ▼❛♣♣✐♥❣ ❤ : ❱ ( ● ) → ❱ ( ❍ ) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝② ✿ ①② ∈ ❊ ( ● ) = ⇒ ❤ ( ① ) ❤ ( ② ) ∈ ❊ ( ❍ ) ◆♦t❛t✐♦♥✿ ● → ❍ ✳ ❚❛r❣❡t ❣r❛♣❤✿ ❍ = ❈ ✺ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✺ ✴ ✸✽

●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ▼❛♣♣✐♥❣ ❤ : ❱ ( ● ) → ❱ ( ❍ ) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝② ✿ ①② ∈ ❊ ( ● ) = ⇒ ❤ ( ① ) ❤ ( ② ) ∈ ❊ ( ❍ ) ◆♦t❛t✐♦♥✿ ● → ❍ ✳ ❚❛r❣❡t ❣r❛♣❤✿ ❍ = ❈ ✺ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✺ ✴ ✸✽

●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ▼❛♣♣✐♥❣ ❤ : ❱ ( ● ) → ❱ ( ❍ ) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝② ✿ ①② ∈ ❊ ( ● ) = ⇒ ❤ ( ① ) ❤ ( ② ) ∈ ❊ ( ❍ ) ◆♦t❛t✐♦♥✿ ● → ❍ ✳ ❚❛r❣❡t ❣r❛♣❤✿ ❍ = ❈ ✺ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✺ ✴ ✸✽

●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ▼❛♣♣✐♥❣ ❤ : ❱ ( ● ) → ❱ ( ❍ ) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝② ✿ ①② ∈ ❊ ( ● ) = ⇒ ❤ ( ① ) ❤ ( ② ) ∈ ❊ ( ❍ ) ◆♦t❛t✐♦♥✿ ● → ❍ ✳ ❈♦♠♣❧❡t❡ ❣r❛♣❤ ❑ ✻ ❘❡♠❛r❦✿ ❍♦♠♦♠♦r♣❤✐s♠s ❣❡♥❡r❛❧✐③❡ ♣r♦♣❡r ❝♦❧♦✉r✐♥❣s ● → ❑ ❦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ χ ( ● ) ≤ ❦ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✺ ✴ ✸✽

❖❞❞ ❝②❝❧❡s Pr♦♣♦s✐t✐♦♥ ❈ ✷ ❦ + ✶ → ❈ ✷ ℓ + ✶ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ℓ ≤ ❦ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✻ ✴ ✸✽

Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❝♦r❡ ♦❢ ❛ ❣r❛♣❤ ✐s ✉♥✐q✉❡ ✭✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✮ ❊①❛♠♣❧❡s✿ t❤❡ ❝♦r❡ ♦❢ ❛♥② ♥♦♥tr✐✈✐❛❧ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✐s ❑ ✷ ❝♦♠♣❧❡t❡ ❣r❛♣❤s ❛♥❞ ♦❞❞ ❝②❝❧❡s ❛r❡ ❝♦r❡s Pr♦♣♦s✐t✐♦♥ ● ❍ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❝♦r❡ ● ❝♦r❡ ❍ ❈♦r❡s ❉❡✜♥✐t✐♦♥ ✲ ❈♦r❡ • ❈♦r❡ ♦❢ ● ✿ ♠✐♥✐♠❛❧ s✉❜❣r❛♣❤ ❍ ✇✐t❤ ● → ❍ • ● ✐s ❛ ❝♦r❡ ✐❢ ❝♦r❡ ( ● ) = ● ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✼ ✴ ✸✽

Pr♦♣♦s✐t✐♦♥ ● ❍ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❝♦r❡ ● ❝♦r❡ ❍ ❈♦r❡s ❉❡✜♥✐t✐♦♥ ✲ ❈♦r❡ • ❈♦r❡ ♦❢ ● ✿ ♠✐♥✐♠❛❧ s✉❜❣r❛♣❤ ❍ ✇✐t❤ ● → ❍ • ● ✐s ❛ ❝♦r❡ ✐❢ ❝♦r❡ ( ● ) = ● Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❝♦r❡ ♦❢ ❛ ❣r❛♣❤ ✐s ✉♥✐q✉❡ ✭✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✮ ❊①❛♠♣❧❡s✿ • t❤❡ ❝♦r❡ ♦❢ ❛♥② ♥♦♥tr✐✈✐❛❧ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✐s ❑ ✷ • ❝♦♠♣❧❡t❡ ❣r❛♣❤s ❛♥❞ ♦❞❞ ❝②❝❧❡s ❛r❡ ❝♦r❡s ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✼ ✴ ✸✽

❈♦r❡s ❉❡✜♥✐t✐♦♥ ✲ ❈♦r❡ • ❈♦r❡ ♦❢ ● ✿ ♠✐♥✐♠❛❧ s✉❜❣r❛♣❤ ❍ ✇✐t❤ ● → ❍ • ● ✐s ❛ ❝♦r❡ ✐❢ ❝♦r❡ ( ● ) = ● Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❝♦r❡ ♦❢ ❛ ❣r❛♣❤ ✐s ✉♥✐q✉❡ ✭✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✮ ❊①❛♠♣❧❡s✿ • t❤❡ ❝♦r❡ ♦❢ ❛♥② ♥♦♥tr✐✈✐❛❧ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✐s ❑ ✷ • ❝♦♠♣❧❡t❡ ❣r❛♣❤s ❛♥❞ ♦❞❞ ❝②❝❧❡s ❛r❡ ❝♦r❡s Pr♦♣♦s✐t✐♦♥ ● → ❍ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❝♦r❡ ( ● ) → ❝♦r❡ ( ❍ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✼ ✴ ✸✽

❈♦r❡s ❉❡✜♥✐t✐♦♥ ✲ ❈♦r❡ • ❈♦r❡ ♦❢ ● ✿ ♠✐♥✐♠❛❧ s✉❜❣r❛♣❤ ❍ ✇✐t❤ ● → ❍ • ● ✐s ❛ ❝♦r❡ ✐❢ ❝♦r❡ ( ● ) = ● Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❝♦r❡ ♦❢ ❛ ❣r❛♣❤ ✐s ✉♥✐q✉❡ ✭✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✮ ❊①❛♠♣❧❡s✿ • t❤❡ ❝♦r❡ ♦❢ ❛♥② ♥♦♥tr✐✈✐❛❧ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✐s ❑ ✷ • ❝♦♠♣❧❡t❡ ❣r❛♣❤s ❛♥❞ ♦❞❞ ❝②❝❧❡s ❛r❡ ❝♦r❡s Pr♦♣♦s✐t✐♦♥ ● → ❍ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❝♦r❡ ( ● ) → ❝♦r❡ ( ❍ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✼ ✴ ✸✽

❚❤❡ ❤♦♠♦♠♦r♣❤✐s♠ ♦r❞❡r ❉❡✜♥✐t✐♦♥ ✲ ❍♦♠♦♠♦r♣❤✐s♠ q✉❛s✐✲♦r❞❡r ❉❡✜♥❡❞ ❜② ● � ❍ ✐✛ ● → ❍ ✭✐❢ r❡str✐❝t❡❞ t♦ ❝♦r❡s✿ ♣❛rt✐❛❧ ♦r❞❡r✮✳ • r❡✢❡①✐✈❡ • tr❛♥s✐t✐✈❡ • ❛♥t✐s②♠♠❡tr✐❝ ✭❝♦r❡s✮ ... ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✽ ✴ ✸✽

❇♦✉♥❞s ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✾ ✴ ✸✽

❇♦✉♥❞s ✐♥ t❤❡ ❤♦♠♦♠♦r♣❤✐s♠ ♦r❞❡r ❉❡✜♥✐t✐♦♥ ✲ ❇♦✉♥❞ ✐♥ t❤❡ ♦r❞❡r ●r❛♣❤ ❇ ✐s ❛ ❜♦✉♥❞ ❢♦r ❣r❛♣❤ ❝❧❛ss C ✐❢ ❢♦r ❡❛❝❤ ● ∈ C ✱ ● → ❇ ✳ ... ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✵ ✴ ✸✽

❇♦✉♥❞s ✐♥ t❤❡ ❤♦♠♦♠♦r♣❤✐s♠ ♦r❞❡r ❉❡✜♥✐t✐♦♥ ✲ ❇♦✉♥❞ ✐♥ t❤❡ ♦r❞❡r ●r❛♣❤ ❇ ✐s ❛ ❜♦✉♥❞ ❢♦r ❣r❛♣❤ ❝❧❛ss C ✐❢ ❢♦r ❡❛❝❤ ● ∈ C ✱ ● → ❇ ✳ ... ❑ ✹ ✿ ❜♦✉♥❞ ❢♦r ♣❧❛♥❛r ❣r❛♣❤s ✭✹❈❚✮ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✵ ✴ ✸✽

❇♦✉♥❞s ✐♥ t❤❡ ❤♦♠♦♠♦r♣❤✐s♠ ♦r❞❡r ❉❡✜♥✐t✐♦♥ ✲ ❇♦✉♥❞ ✐♥ t❤❡ ♦r❞❡r ●r❛♣❤ ❇ ✐s ❛ ❜♦✉♥❞ ❢♦r ❣r❛♣❤ ❝❧❛ss C ✐❢ ❢♦r ❡❛❝❤ ● ∈ C ✱ ● → ❇ ✳ ... ❑ ✸ ✿ ❜♦✉♥❞ ❢♦r ♣❧❛♥❛r tr✐❛♥❣❧❡✲❢r❡❡ ❣r❛♣❤s ✭●röt③s❝❤✬s t❤❡♦r❡♠✮ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✵ ✴ ✸✽

❉❡✜♥✐t✐♦♥ ✿ ✜♥✐t❡ s❡t ♦❢ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳ ❋♦r❜ ✿ ❣r❛♣❤s ● s✳t✳ ❢♦r ❛♥② ❋ ✱ ● ✳ ❋ ❊①❛♠♣❧❡s✿ ✿ ❣r❛♣❤s ✇✐t❤ ❝❧✐q✉❡ ♥✉♠❜❡r ❛t ♠♦st ✶ ❋♦r❜ ❑ ❋♦r❜ ❈ ✷ ❦ ✿ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ ✶ ✶ ✭♦❞❞✲❣✐rt❤✿ ❧❡♥❣t❤ ♦❢ ❛ s♠❛❧❧❡st ♦❞❞ ❝②❝❧❡✮ ❚❤❡♦r❡♠ ✭❍ä❣❣✈✐st✲❍❡❧❧✱ ✶✾✾✸✮ ❆❧❧ ❦ ✲❝♦❧♦r❛❜❧❡ ❣r❛♣❤s ♦❢ ❋♦r❜ ✇✐t❤ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ❞ ❛r❡ ❜♦✉♥❞❡❞ ❜② ❛ ❦ ✲❝♦❧♦r❛❜❧❡ ❣r❛♣❤ ❇ ❦ ❞ ✐♥ ❋♦r❜ ✳ ❆ ❣❡♥❡r✐❝ ♣r♦❜❧❡♠ ◗✉❡st✐♦♥ ●✐✈❡♥ ❣r❛♣❤ ❝❧❛ss C ✱ ✐s t❤❡r❡ ❛ ❜♦✉♥❞ ❢♦r C ❤❛✈✐♥❣ s♣❡❝✐✜❝ ♣r♦♣❡rt✐❡s❄ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✶ ✴ ✸✽

❚❤❡♦r❡♠ ✭❍ä❣❣✈✐st✲❍❡❧❧✱ ✶✾✾✸✮ ❆❧❧ ❦ ✲❝♦❧♦r❛❜❧❡ ❣r❛♣❤s ♦❢ ❋♦r❜ ✇✐t❤ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ❞ ❛r❡ ❜♦✉♥❞❡❞ ❜② ❛ ❦ ✲❝♦❧♦r❛❜❧❡ ❣r❛♣❤ ❇ ❦ ❞ ✐♥ ❋♦r❜ ✳ ❆ ❣❡♥❡r✐❝ ♣r♦❜❧❡♠ ◗✉❡st✐♦♥ ●✐✈❡♥ ❣r❛♣❤ ❝❧❛ss C ✱ ✐s t❤❡r❡ ❛ ❜♦✉♥❞ ❢♦r C ❤❛✈✐♥❣ s♣❡❝✐✜❝ ♣r♦♣❡rt✐❡s❄ ❉❡✜♥✐t✐♦♥ F ✿ ✜♥✐t❡ s❡t ♦❢ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳ ❋♦r❜ ( F ) ✿ ❣r❛♣❤s ● s✳t✳ ❢♦r ❛♥② ❋ ∈ F ✱ ❋ �→ ● ✳ ❊①❛♠♣❧❡s✿ • ❋♦r❜ ( { ❑ ℓ } ) ✿ ❣r❛♣❤s ✇✐t❤ ❝❧✐q✉❡ ♥✉♠❜❡r ❛t ♠♦st ℓ − ✶ • ❋♦r❜ ( { ❈ ✷ ❦ − ✶ } ) ✿ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶ ✭♦❞❞✲❣✐rt❤✿ ❧❡♥❣t❤ ♦❢ ❛ s♠❛❧❧❡st ♦❞❞ ❝②❝❧❡✮ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✶ ✴ ✸✽

❆ ❣❡♥❡r✐❝ ♣r♦❜❧❡♠ ◗✉❡st✐♦♥ ●✐✈❡♥ ❣r❛♣❤ ❝❧❛ss C ✱ ✐s t❤❡r❡ ❛ ❜♦✉♥❞ ❢♦r C ❤❛✈✐♥❣ s♣❡❝✐✜❝ ♣r♦♣❡rt✐❡s❄ ❉❡✜♥✐t✐♦♥ F ✿ ✜♥✐t❡ s❡t ♦❢ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳ ❋♦r❜ ( F ) ✿ ❣r❛♣❤s ● s✳t✳ ❢♦r ❛♥② ❋ ∈ F ✱ ❋ �→ ● ✳ ❊①❛♠♣❧❡s✿ • ❋♦r❜ ( { ❑ ℓ } ) ✿ ❣r❛♣❤s ✇✐t❤ ❝❧✐q✉❡ ♥✉♠❜❡r ❛t ♠♦st ℓ − ✶ • ❋♦r❜ ( { ❈ ✷ ❦ − ✶ } ) ✿ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶ ✭♦❞❞✲❣✐rt❤✿ ❧❡♥❣t❤ ♦❢ ❛ s♠❛❧❧❡st ♦❞❞ ❝②❝❧❡✮ ❚❤❡♦r❡♠ ✭❍ä❣❣✈✐st✲❍❡❧❧✱ ✶✾✾✸✮ ❆❧❧ ❦ ✲❝♦❧♦r❛❜❧❡ ❣r❛♣❤s ♦❢ ❋♦r❜ ( F ) ✇✐t❤ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ❞ ❛r❡ ❜♦✉♥❞❡❞ ❜② ❛ ❦ ✲❝♦❧♦r❛❜❧❡ ❣r❛♣❤ ❇ ( ❦ , ❞ , F ) ✐♥ ❋♦r❜ ( F ) ✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✶ ✴ ✸✽

◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✸✿ ✸✲❝♦❧♦r❛❜❧❡ ❑ ✸ ✲❢r❡❡ ❜♦✉♥❞ ❢♦r ❑ ✸ ✲❢r❡❡ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦r❞❡r r♦✉❣❤❧② ✶✵ ✶✵ ✷✷✺ ✭✐♥ ❣❡♥❡r❛❧✱ ❢♦r ❛♥② ♠✐♥♦r✲❝❧♦s❡❞ ❝❧❛ss✮ ◆❛s❡r❛sr✱ ✷✵✵✻✿ ❑ ✺ ✲❢r❡❡ ❜♦✉♥❞ ❢♦r ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦r❞❡r ✻✸ ✷ ✷ ✻✷ ✶✵ ✸✽✾✺✾✸✸ ✺ ◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✻✿ ❑ ❦ ✲❢r❡❡ ❜♦✉♥❞s ✭❢♦r ❑ ❦ ✲❢r❡❡ ❣r❛♣❤s ❢r♦♠ ❛♥② ♠✐♥♦r✲❝❧♦s❡❞ ❝❧❛ss✮ ❆ ❣❡♥❡r✐❝ ♣r♦❜❧❡♠ ❋♦✉r ❈♦❧♦r ❚❤❡♦r❡♠✿ ✭ ❑ ✺ ✲❢r❡❡✮ ♣❧❛♥❛r ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✺ ✲❢r❡❡ ❣r❛♣❤ ✭ ❑ ✹ ✮ ●röt③s❝❤✬s ❚❤❡♦r❡♠✿ ❑ ✸ ✲❢r❡❡ ♣❧❛♥❛r ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❑ ✸ ◗✉❡st✐♦♥ ✭◆❡➨❡t➦✐❧✱ ✶✾✾✾✮ • ❆r❡ ♣❧❛♥❛r ❑ ✸ ✲❢r❡❡ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✸ ✲❢r❡❡ ❣r❛♣❤❄ • ❆r❡ ♣❧❛♥❛r ❑ ✹ ✲❢r❡❡ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✹ ✲❢r❡❡ ❣r❛♣❤❄ • ❆r❡ ♣❧❛♥❛r ✭ ❑ ✺ ✲❢r❡❡✮ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✺ ✲❢r❡❡ ❣r❛♣❤❄ ▼✐♥♦r ♦❢ ● ✿ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❜② s❡q✉❡♥❝❡ ♦❢ ❡❞❣❡✲❝♦♥tr❛❝t✐♦♥s ❛♥❞ ❞❡❧❡t✐♦♥s✳ ❈❧❛ss✐❝ ♠✐♥♦r✲❝❧♦s❡❞ ❣r❛♣❤ ❝❧❛ss❡s✿ tr❡❡s✱ ♣❧❛♥❛r ❣r❛♣❤s✱ ❜♦✉♥❞❡❞ ❣❡♥✉s✱ ❝❧❛ss❡❞ ❞❡✜♥❡❞ ❜② ❢♦r❜✐❞❞❡♥ ♠✐♥♦r✳✳✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✷ ✴ ✸✽

◆❛s❡r❛sr✱ ✷✵✵✻✿ ❑ ✺ ✲❢r❡❡ ❜♦✉♥❞ ❢♦r ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦r❞❡r ✻✸ ✷ ✷ ✻✷ ✶✵ ✸✽✾✺✾✸✸ ✺ ◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✻✿ ❑ ❦ ✲❢r❡❡ ❜♦✉♥❞s ✭❢♦r ❑ ❦ ✲❢r❡❡ ❣r❛♣❤s ❢r♦♠ ❛♥② ♠✐♥♦r✲❝❧♦s❡❞ ❝❧❛ss✮ ❆ ❣❡♥❡r✐❝ ♣r♦❜❧❡♠ ❋♦✉r ❈♦❧♦r ❚❤❡♦r❡♠✿ ✭ ❑ ✺ ✲❢r❡❡✮ ♣❧❛♥❛r ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✺ ✲❢r❡❡ ❣r❛♣❤ ✭ ❑ ✹ ✮ ●röt③s❝❤✬s ❚❤❡♦r❡♠✿ ❑ ✸ ✲❢r❡❡ ♣❧❛♥❛r ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❑ ✸ ◗✉❡st✐♦♥ ✭◆❡➨❡t➦✐❧✱ ✶✾✾✾✮ • ❆r❡ ♣❧❛♥❛r ❑ ✸ ✲❢r❡❡ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✸ ✲❢r❡❡ ❣r❛♣❤❄ • ❆r❡ ♣❧❛♥❛r ❑ ✹ ✲❢r❡❡ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✹ ✲❢r❡❡ ❣r❛♣❤❄ • ❆r❡ ♣❧❛♥❛r ✭ ❑ ✺ ✲❢r❡❡✮ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✺ ✲❢r❡❡ ❣r❛♣❤❄ ▼✐♥♦r ♦❢ ● ✿ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❜② s❡q✉❡♥❝❡ ♦❢ ❡❞❣❡✲❝♦♥tr❛❝t✐♦♥s ❛♥❞ ❞❡❧❡t✐♦♥s✳ ❈❧❛ss✐❝ ♠✐♥♦r✲❝❧♦s❡❞ ❣r❛♣❤ ❝❧❛ss❡s✿ tr❡❡s✱ ♣❧❛♥❛r ❣r❛♣❤s✱ ❜♦✉♥❞❡❞ ❣❡♥✉s✱ ❝❧❛ss❡❞ ❞❡✜♥❡❞ ❜② ❢♦r❜✐❞❞❡♥ ♠✐♥♦r✳✳✳ • ◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✸✿ ✸✲❝♦❧♦r❛❜❧❡ ❑ ✸ ✲❢r❡❡ ❜♦✉♥❞ ❢♦r ❑ ✸ ✲❢r❡❡ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦r❞❡r r♦✉❣❤❧② ✶✵ ✶✵ ✷✷✺ ✭✐♥ ❣❡♥❡r❛❧✱ ❢♦r ❛♥② ♠✐♥♦r✲❝❧♦s❡❞ ❝❧❛ss✮ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✷ ✴ ✸✽

◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✻✿ ❑ ❦ ✲❢r❡❡ ❜♦✉♥❞s ✭❢♦r ❑ ❦ ✲❢r❡❡ ❣r❛♣❤s ❢r♦♠ ❛♥② ♠✐♥♦r✲❝❧♦s❡❞ ❝❧❛ss✮ ❆ ❣❡♥❡r✐❝ ♣r♦❜❧❡♠ ❋♦✉r ❈♦❧♦r ❚❤❡♦r❡♠✿ ✭ ❑ ✺ ✲❢r❡❡✮ ♣❧❛♥❛r ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✺ ✲❢r❡❡ ❣r❛♣❤ ✭ ❑ ✹ ✮ ●röt③s❝❤✬s ❚❤❡♦r❡♠✿ ❑ ✸ ✲❢r❡❡ ♣❧❛♥❛r ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❑ ✸ ◗✉❡st✐♦♥ ✭◆❡➨❡t➦✐❧✱ ✶✾✾✾✮ • ❆r❡ ♣❧❛♥❛r ❑ ✸ ✲❢r❡❡ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✸ ✲❢r❡❡ ❣r❛♣❤❄ • ❆r❡ ♣❧❛♥❛r ❑ ✹ ✲❢r❡❡ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✹ ✲❢r❡❡ ❣r❛♣❤❄ • ❆r❡ ♣❧❛♥❛r ✭ ❑ ✺ ✲❢r❡❡✮ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✺ ✲❢r❡❡ ❣r❛♣❤❄ ▼✐♥♦r ♦❢ ● ✿ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❜② s❡q✉❡♥❝❡ ♦❢ ❡❞❣❡✲❝♦♥tr❛❝t✐♦♥s ❛♥❞ ❞❡❧❡t✐♦♥s✳ ❈❧❛ss✐❝ ♠✐♥♦r✲❝❧♦s❡❞ ❣r❛♣❤ ❝❧❛ss❡s✿ tr❡❡s✱ ♣❧❛♥❛r ❣r❛♣❤s✱ ❜♦✉♥❞❡❞ ❣❡♥✉s✱ ❝❧❛ss❡❞ ❞❡✜♥❡❞ ❜② ❢♦r❜✐❞❞❡♥ ♠✐♥♦r✳✳✳ • ◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✸✿ ✸✲❝♦❧♦r❛❜❧❡ ❑ ✸ ✲❢r❡❡ ❜♦✉♥❞ ❢♦r ❑ ✸ ✲❢r❡❡ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦r❞❡r r♦✉❣❤❧② ✶✵ ✶✵ ✷✷✺ ✭✐♥ ❣❡♥❡r❛❧✱ ❢♦r ❛♥② ♠✐♥♦r✲❝❧♦s❡❞ ❝❧❛ss✮ • ◆❛s❡r❛sr✱ ✷✵✵✻✿ ❑ ✺ ✲❢r❡❡ ❜♦✉♥❞ ❢♦r ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦r❞❡r ✻✸ · ✷ ✷ ( ✻✷ ✺ ) ≈ ✶✵ ✸✽✾✺✾✸✸ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✷ ✴ ✸✽

❆ ❣❡♥❡r✐❝ ♣r♦❜❧❡♠ ❋♦✉r ❈♦❧♦r ❚❤❡♦r❡♠✿ ✭ ❑ ✺ ✲❢r❡❡✮ ♣❧❛♥❛r ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✺ ✲❢r❡❡ ❣r❛♣❤ ✭ ❑ ✹ ✮ ●röt③s❝❤✬s ❚❤❡♦r❡♠✿ ❑ ✸ ✲❢r❡❡ ♣❧❛♥❛r ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❑ ✸ ◗✉❡st✐♦♥ ✭◆❡➨❡t➦✐❧✱ ✶✾✾✾✮ • ❆r❡ ♣❧❛♥❛r ❑ ✸ ✲❢r❡❡ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✸ ✲❢r❡❡ ❣r❛♣❤❄ • ❆r❡ ♣❧❛♥❛r ❑ ✹ ✲❢r❡❡ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✹ ✲❢r❡❡ ❣r❛♣❤❄ • ❆r❡ ♣❧❛♥❛r ✭ ❑ ✺ ✲❢r❡❡✮ ❣r❛♣❤s ❜♦✉♥❞❡❞ ❜② ❛ ❑ ✺ ✲❢r❡❡ ❣r❛♣❤❄ ▼✐♥♦r ♦❢ ● ✿ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❜② s❡q✉❡♥❝❡ ♦❢ ❡❞❣❡✲❝♦♥tr❛❝t✐♦♥s ❛♥❞ ❞❡❧❡t✐♦♥s✳ ❈❧❛ss✐❝ ♠✐♥♦r✲❝❧♦s❡❞ ❣r❛♣❤ ❝❧❛ss❡s✿ tr❡❡s✱ ♣❧❛♥❛r ❣r❛♣❤s✱ ❜♦✉♥❞❡❞ ❣❡♥✉s✱ ❝❧❛ss❡❞ ❞❡✜♥❡❞ ❜② ❢♦r❜✐❞❞❡♥ ♠✐♥♦r✳✳✳ • ◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✸✿ ✸✲❝♦❧♦r❛❜❧❡ ❑ ✸ ✲❢r❡❡ ❜♦✉♥❞ ❢♦r ❑ ✸ ✲❢r❡❡ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦r❞❡r r♦✉❣❤❧② ✶✵ ✶✵ ✷✷✺ ✭✐♥ ❣❡♥❡r❛❧✱ ❢♦r ❛♥② ♠✐♥♦r✲❝❧♦s❡❞ ❝❧❛ss✮ • ◆❛s❡r❛sr✱ ✷✵✵✻✿ ❑ ✺ ✲❢r❡❡ ❜♦✉♥❞ ❢♦r ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦r❞❡r ✻✸ · ✷ ✷ ( ✻✷ ✺ ) ≈ ✶✵ ✸✽✾✺✾✸✸ • ◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✻✿ ❑ ❦ ✲❢r❡❡ ❜♦✉♥❞s ✭❢♦r ❑ ❦ ✲❢r❡❡ ❣r❛♣❤s ❢r♦♠ ❛♥② ♠✐♥♦r✲❝❧♦s❡❞ ❝❧❛ss✮ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✷ ✴ ✸✽

◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③ t❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥ F ✿ ✜♥✐t❡ s❡t ♦❢ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳ ❋♦r❜ ( F ) ✿ ❛❧❧ ❣r❛♣❤s ● s✳t✳ ❢♦r ❛♥② ❋ ∈ F ✱ ❋ �→ ● ✳ ❚❤❡♦r❡♠ ✭◆❡➨❡t➦✐❧ ❛♥❞ ❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✽✮ ❋♦r ❛♥② ♠✐♥♦r✲❝❧♦s❡❞ ❝❧❛ss C ♦❢ ❣r❛♣❤s✿ C ∩ ❋♦r❜ ( F ) ✐s ❜♦✉♥❞❡❞ ❜② ❛ ✜♥✐t❡ ❣r❛♣❤ ❇ ( C , F ) ❢r♦♠ ❋♦r❜ ( F ) ✳ B ( C , F ) Forb ( F ) C ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✸ ✴ ✸✽

◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③ t❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥ F ✿ ✜♥✐t❡ s❡t ♦❢ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳ ❋♦r❜ ( F ) ✿ ❛❧❧ ❣r❛♣❤s ● s✳t✳ ❢♦r ❛♥② ❋ ∈ F ✱ ❋ �→ ● ✳ ❚❤❡♦r❡♠ ✭◆❡➨❡t➦✐❧ ❛♥❞ ❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✽✮ ❋♦r ❛♥② ❜♦✉♥❞❡❞ ❡①♣❛♥s✐♦♥ ❝❧❛ss C ♦❢ ❣r❛♣❤s✿ C ∩ ❋♦r❜ ( F ) ✐s ❜♦✉♥❞❡❞ ❜② ❛ ✜♥✐t❡ ❣r❛♣❤ ❇ ( C , F ) ❢r♦♠ ❋♦r❜ ( F ) ✳ Pr♦✈❡❞ ✉s✐♥❣ ♠❛❝❤✐♥❡r② ♦❢ t❤❡ B ( C , F ) s♣❛rs✐❝✐t② ♣r♦❥❡❝t Forb ( F ) C ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✸ ✴ ✸✽

◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③ t❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥ F ✿ ✜♥✐t❡ s❡t ♦❢ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳ ❋♦r❜ ( F ) ✿ ❛❧❧ ❣r❛♣❤s ● s✳t✳ ❢♦r ❛♥② ❋ ∈ F ✱ ❋ �→ ● ✳ ❚❤❡♦r❡♠ ✭◆❡➨❡t➦✐❧ ❛♥❞ ❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✽✮ ❋♦r ❛♥② ❜♦✉♥❞❡❞ ❡①♣❛♥s✐♦♥ ❝❧❛ss C ♦❢ ❣r❛♣❤s✿ C ∩ ❋♦r❜ ( F ) ✐s ❜♦✉♥❞❡❞ ❜② ❛ ✜♥✐t❡ ❣r❛♣❤ ❇ ( C , F ) ❢r♦♠ ❋♦r❜ ( F ) ✳ ❊①❛♠♣❧❡ ✶✳ C ✿ ♣❧❛♥❛r ❣r❛♣❤s B ( C , F ) F = { ❈ ✷ ❦ − ✶ } − → ❛❧❧ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st Forb ( F ) ✷ ❦ + ✶ ♠❛♣ t♦ s♦♠❡ ❣r❛♣❤ ❇ ♥ , ❦ ♦❢ C ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✸ ✴ ✸✽

◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③ t❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥ F ✿ ✜♥✐t❡ s❡t ♦❢ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳ ❋♦r❜ ( F ) ✿ ❛❧❧ ❣r❛♣❤s ● s✳t✳ ❢♦r ❛♥② ❋ ∈ F ✱ ❋ �→ ● ✳ ❚❤❡♦r❡♠ ✭◆❡➨❡t➦✐❧ ❛♥❞ ❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✽✮ ❋♦r ❛♥② ❜♦✉♥❞❡❞ ❡①♣❛♥s✐♦♥ ❝❧❛ss C ♦❢ ❣r❛♣❤s✿ C ∩ ❋♦r❜ ( F ) ✐s ❜♦✉♥❞❡❞ ❜② ❛ ✜♥✐t❡ ❣r❛♣❤ ❇ ( C , F ) ❢r♦♠ ❋♦r❜ ( F ) ✳ ❊①❛♠♣❧❡ ✷✳ C ✿ ❑ ♥ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s B ( C , F ) F = { ❑ ♥ } − → ❛❧❧ ❑ ♥ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♠❛♣ t♦ s♦♠❡ Forb ( F ) ❣r❛♣❤ ❇ ♥ ♦❢ ❝❧✐q✉❡ ♥✉♠❜❡r ♥ − ✶✳ C ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✸ ✴ ✸✽

◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③ t❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥ F ✿ ✜♥✐t❡ s❡t ♦❢ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳ ❋♦r❜ ( F ) ✿ ❛❧❧ ❣r❛♣❤s ● s✳t✳ ❢♦r ❛♥② ❋ ∈ F ✱ ❋ �→ ● ✳ ❚❤❡♦r❡♠ ✭◆❡➨❡t➦✐❧ ❛♥❞ ❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✽✮ ❋♦r ❛♥② ❜♦✉♥❞❡❞ ❡①♣❛♥s✐♦♥ ❝❧❛ss C ♦❢ ❣r❛♣❤s✿ C ∩ ❋♦r❜ ( F ) ✐s ❜♦✉♥❞❡❞ ❜② ❛ ✜♥✐t❡ ❣r❛♣❤ ❇ ( C , F ) ❢r♦♠ ❋♦r❜ ( F ) ✳ ◆♦t❡✿ t❤❡r❡ ❝♦✉❧❞ ❜❡ ♥♦ ❜♦✉♥❞ ✐♥ B ( C , F ) C ∩ ❋♦r❜ ( F ) ✐ts❡❧❢✦ ✭❡✳❣✳ ♣❧❛♥❛r tr✐❛♥❣❧❡✲❢r❡❡ ❣r❛♣❤s✱ ◆❛s❡r❛sr ✷✵✵✺✮ Forb ( F ) C ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✸ ✴ ✸✽

◆❡➨❡t➦✐❧✲❖ss♦♥❛ ❞❡ ▼❡♥❞❡③ t❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥ F ✿ ✜♥✐t❡ s❡t ♦❢ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳ ❋♦r❜ ( F ) ✿ ❛❧❧ ❣r❛♣❤s ● s✳t✳ ❢♦r ❛♥② ❋ ∈ F ✱ ❋ �→ ● ✳ ❚❤❡♦r❡♠ ✭◆❡➨❡t➦✐❧ ❛♥❞ ❖ss♦♥❛ ❞❡ ▼❡♥❞❡③✱ ✷✵✵✽✮ ❋♦r ❛♥② ❜♦✉♥❞❡❞ ❡①♣❛♥s✐♦♥ ❝❧❛ss C ♦❢ ❣r❛♣❤s✿ C ∩ ❋♦r❜ ( F ) ✐s ❜♦✉♥❞❡❞ ❜② ❛ ✜♥✐t❡ ❣r❛♣❤ ❇ ( C , F ) ❢r♦♠ ❋♦r❜ ( F ) ✳ ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛ ❜♦✉♥❞ ♦❢ s♠❛❧❧❡st ♦r❞❡r❄ ❊①❛♠♣❧❡✿ C ✿ ❑ ♥ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s✱ F = { ❑ ♥ } − → ❍❛❞✇✐❣❡r✬s ❝♦♥❥❡❝t✉r❡ st❛t❡s t❤❛t s♠❛❧❧❡st ❇ ♥ ✐s ❑ ♥ − ✶ ✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✸ ✴ ✸✽

Pr♦❥❡❝t✐✈❡ ❝✉❜❡s ❛♥❞ ♣❧❛♥❛r ❣r❛♣❤s ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✹ ✴ ✸✽

P❈ ✹ ✿ ❈❧❡❜s❝❤ ❣r❛♣❤ ✭❛✳❦✳❛ ●r❡❡♥✇♦♦❞✲●❧❡❛s♦♥✿ ❘ ✸ ✸ ✸ ✶✼✱ ✶✾✺✺✮ ◆❛s❡r❛sr✬s ❝♦♥❥❡❝t✉r❡ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✺ ✴ ✸✽

P❈ ✹ ✿ ❈❧❡❜s❝❤ ❣r❛♣❤ ✭❛✳❦✳❛ ●r❡❡♥✇♦♦❞✲●❧❡❛s♦♥✿ ❘ ✸ ✸ ✸ ✶✼✱ ✶✾✺✺✮ ◆❛s❡r❛sr✬s ❝♦♥❥❡❝t✉r❡ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❉❡✜♥✐t✐♦♥ ✲ Pr♦❥❡❝t✐✈❡ ❝✉❜❡ ♦❢ ❞✐♠❡♥s✐♦♥ ❞ ✱ P❈ ( ❞ ) ❖❜t❛✐♥❡❞ ❢r♦♠ ❤②♣❡r❝✉❜❡ ❍ ( ❞ ) ❜② ❛❞❞✐♥❣ ❡❞❣❡s ❜❡t✇❡❡♥ ❛❧❧ ❛♥t✐♣♦❞❛❧ ♣❛✐rs✳ ❆❧s♦ ❦♥♦✇♥ ❛s ❢♦❧❞❡❞ ❝✉❜❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✺ ✴ ✸✽

P❈ ✹ ✿ ❈❧❡❜s❝❤ ❣r❛♣❤ ✭❛✳❦✳❛ ●r❡❡♥✇♦♦❞✲●❧❡❛s♦♥✿ ❘ ✸ ✸ ✸ ✶✼✱ ✶✾✺✺✮ ◆❛s❡r❛sr✬s ❝♦♥❥❡❝t✉r❡ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❉❡✜♥✐t✐♦♥ ✲ Pr♦❥❡❝t✐✈❡ ❝✉❜❡ ♦❢ ❞✐♠❡♥s✐♦♥ ❞ ✱ P❈ ( ❞ ) ❖❜t❛✐♥❡❞ ❢r♦♠ ❤②♣❡r❝✉❜❡ ❍ ( ❞ ) ❜② ❛❞❞✐♥❣ ❡❞❣❡s ❜❡t✇❡❡♥ ❛❧❧ ❛♥t✐♣♦❞❛❧ ♣❛✐rs✳ ❆❧s♦ ❦♥♦✇♥ ❛s ❢♦❧❞❡❞ ❝✉❜❡✳ ❍ ( ✷ ) ❍ ( ✸ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✺ ✴ ✸✽

P❈ ✹ ✿ ❈❧❡❜s❝❤ ❣r❛♣❤ ✭❛✳❦✳❛ ●r❡❡♥✇♦♦❞✲●❧❡❛s♦♥✿ ❘ ✸ ✸ ✸ ✶✼✱ ✶✾✺✺✮ ◆❛s❡r❛sr✬s ❝♦♥❥❡❝t✉r❡ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❉❡✜♥✐t✐♦♥ ✲ Pr♦❥❡❝t✐✈❡ ❝✉❜❡ ♦❢ ❞✐♠❡♥s✐♦♥ ❞ ✱ P❈ ( ❞ ) ❖❜t❛✐♥❡❞ ❢r♦♠ ❤②♣❡r❝✉❜❡ ❍ ( ❞ ) ❜② ❛❞❞✐♥❣ ❡❞❣❡s ❜❡t✇❡❡♥ ❛❧❧ ❛♥t✐♣♦❞❛❧ ♣❛✐rs✳ ❆❧s♦ ❦♥♦✇♥ ❛s ❢♦❧❞❡❞ ❝✉❜❡✳ P❈ ( ✷ ) = ❑ ✹ P❈ ( ✸ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✺ ✴ ✸✽

P❈ ✹ ✿ ❈❧❡❜s❝❤ ❣r❛♣❤ ✭❛✳❦✳❛ ●r❡❡♥✇♦♦❞✲●❧❡❛s♦♥✿ ❘ ✸ ✸ ✸ ✶✼✱ ✶✾✺✺✮ ◆❛s❡r❛sr✬s ❝♦♥❥❡❝t✉r❡ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❉❡✜♥✐t✐♦♥ ✲ Pr♦❥❡❝t✐✈❡ ❝✉❜❡ ♦❢ ❞✐♠❡♥s✐♦♥ ❞ ✱ P❈ ( ❞ ) ❖❜t❛✐♥❡❞ ❢r♦♠ ❤②♣❡r❝✉❜❡ ❍ ( ❞ ) ❜② ❛❞❞✐♥❣ ❡❞❣❡s ❜❡t✇❡❡♥ ❛❧❧ ❛♥t✐♣♦❞❛❧ ♣❛✐rs✳ ❆❧s♦ ❦♥♦✇♥ ❛s ❢♦❧❞❡❞ ❝✉❜❡✳ P❈ ( ✷ ) = ❑ ✹ P❈ ( ✸ ) = ❑ ✹ , ✹ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✺ ✴ ✸✽

◆❛s❡r❛sr✬s ❝♦♥❥❡❝t✉r❡ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❉❡✜♥✐t✐♦♥ ✲ Pr♦❥❡❝t✐✈❡ ❝✉❜❡ ♦❢ ❞✐♠❡♥s✐♦♥ ❞ ✱ P❈ ( ❞ ) ❖❜t❛✐♥❡❞ ❢r♦♠ ❤②♣❡r❝✉❜❡ ❍ ( ❞ ) ❜② ❛❞❞✐♥❣ ❡❞❣❡s ❜❡t✇❡❡♥ ❛❧❧ ❛♥t✐♣♦❞❛❧ ♣❛✐rs✳ ❆❧s♦ ❦♥♦✇♥ ❛s ❢♦❧❞❡❞ ❝✉❜❡✳ P❈ ( ✷ ) = ❑ ✹ P❈ ( ✸ ) = ❑ ✹ , ✹ P❈ ( ✹ ) ✿ ❈❧❡❜s❝❤ ❣r❛♣❤ ✭❛✳❦✳❛ ●r❡❡♥✇♦♦❞✲●❧❡❛s♦♥✿ ❘ ( ✸ , ✸ , ✸ ) = ✶✼✱ ✶✾✺✺✮ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✺ ✴ ✸✽

Pr♦❥❡❝t✐✈❡ ❝✉❜❡s ❉❡✜♥✐t✐♦♥ ✲ Pr♦❥❡❝t✐✈❡ ❝✉❜❡ ♦❢ ❞✐♠❡♥s✐♦♥ ❞ ✱ P❈ ( ❞ ) ❖❜t❛✐♥❡❞ ❢r♦♠ ❤②♣❡r❝✉❜❡ ❍ ( ❞ ) ❜② ❛❞❞✐♥❣ ❡❞❣❡s ❜❡t✇❡❡♥ ❛❧❧ ❛♥t✐♣♦❞❛❧ ♣❛✐rs✳ P❈ ( ✷ ) = ❑ ✹ P❈ ( ✸ ) = ❑ ✹ , ✹ P❈ ( ✹ ) ✿ ❈❧❡❜s❝❤ ❣r❛♣❤ ❘❡♠❛r❦ P❈ ( ❞ ) ✐s ❞✐st❛♥❝❡✲tr❛♥s✐t✐✈❡✿ ❢♦r ❛♥② t✇♦ ♣❛✐rs { ① , ② } ✱ { ✉ , ✈ } ✇✐t❤ ❞ ( ① , ② ) = ❞ ( ✉ , ✈ ) ✱ t❤❡r❡ ✐s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ✇✐t❤ ① → ✉ ❛♥❞ ② → ✈ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✻ ✴ ✸✽

Pr♦❥❡❝t✐✈❡ ❝✉❜❡s ❉❡✜♥✐t✐♦♥ ✲ Pr♦❥❡❝t✐✈❡ ❝✉❜❡ ♦❢ ❞✐♠❡♥s✐♦♥ ❞ ✱ P❈ ( ❞ ) ❖❜t❛✐♥❡❞ ❢r♦♠ ❤②♣❡r❝✉❜❡ ❍ ( ❞ ) ❜② ❛❞❞✐♥❣ ❡❞❣❡s ❜❡t✇❡❡♥ ❛❧❧ ❛♥t✐♣♦❞❛❧ ♣❛✐rs✳ P❈ ( ✷ ) = ❑ ✹ P❈ ( ✸ ) = ❑ ✹ , ✹ P❈ ( ✹ ) ✿ ❈❧❡❜s❝❤ ❣r❛♣❤ ❘❡♠❛r❦ ❞ = ✷ ❦ + ✶ ♦❞❞✿ P❈ ( ✷ ❦ + ✶ ) ❜✐♣❛rt✐t❡ ❞ = ✷ ❦ ❡✈❡♥✿ P❈ ( ✷ ❦ ) ❤❛s ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✻ ✴ ✸✽

◆❛s❡r❛sr✬s ❝♦♥❥❡❝t✉r❡ ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❚❤❡♦r❡♠ ✭◆❛s❡r❛sr✱ ❙❡♥✱ ❙✉♥✱ ✷✵✶✹✮ ■❢ tr✉❡✱ t❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ♦♣t✐♠❛❧✿ t❤❡r❡ ✐s ❛ ♣❧❛♥❛r ❣r❛♣❤ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ✇❤♦s❡ s♠❛❧❧❡st ✐♠❛❣❡ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❤❛s ♦r❞❡r ✷ ✷ ❦ ✳ Pr♦♦❢ ✐❞❡❛✿ ❝♦♥str✉❝t ♣❧❛♥❛r ( ✷ ❦ − ✶ ) ✲✇❛❧❦✲♣♦✇❡r ❝❧✐q✉❡ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✼ ✴ ✸✽

◆❛s❡r❛sr ✈s✳ ❙❡②♠♦✉r ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❙❡②♠♦✉r✱ ✶✾✽✶✮ ❊✈❡r② ♣❧❛♥❛r r ✲❣r❛♣❤ ✐s r ✲❡❞❣❡✲❝♦❧♦✉r❛❜❧❡✳ ✭ r ✲❣r❛♣❤✿ r ✲r❡❣✉❧❛r ♠✉❧t✐❣r❛♣❤ ✇✐t❤♦✉t ♦❞❞ ( < r ) ✲❝✉t✮ − → Pr♦✈❡❞ ✉♣ t♦ r = ✽✳ ❚❤❡♦r❡♠ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ P❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶ ❛r❡ ❜♦✉♥❞❡❞ ❜② P❈ ( ✷ ❦ ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ♣❧❛♥❛r ( ✷ ❦ + ✶ ) ✲❣r❛♣❤ ✐s ( ✷ ❦ + ✶ ) ✲❡❞❣❡✲❝♦❧♦✉r❛❜❧❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✽ ✴ ✸✽

◆❛s❡r❛sr✬s ❝♦♥❥❡❝t✉r❡ ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ odd girth 3 5 7 9 ... K 4 K 3 ??? ??? ??? forbidden minor ... K 5 K 4 =PC(2) PC(4) PC(6) PC(8)? Naserasr's conjecture K 6 K 5 K 7 K 6 ? ... Hadwiger's conjecture ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✶✾ ✴ ✸✽

❖✉t❡r♣❧❛♥❛r ❣r❛♣❤s ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✵ ✴ ✸✽

❖✉t❡r♣❧❛♥❛r ❣r❛♣❤s ❖✉t❡r♣❧❛♥❛r ❣r❛♣❤✿ P❧❛♥❛r ❣r❛♣❤s ✇✐t❤ ❛❧❧ ✈❡rt✐❝❡s ♦♥ t❤❡ ♦✉t❡r ❢❛❝❡ − → ❊①❛❝t❧② t❤❡ ❝❧❛ss ♦❢ { ❑ ✹ , ❑ ✷ , ✸ } ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s✳ ❚❤❡♦r❡♠ ✭●❡r❛r❞s✱ ✶✾✽✽✮ ❚❤❡ ❝❧❛ss ♦❢ ♦✉t❡r♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶ ✐s ❜♦✉♥❞❡❞ ❜② t❤❡ ❝②❝❧❡ ❈ ✷ ❦ + ✶ ✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✶ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✷ ✴ ✸✽

Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❘❡♠❛r❦ ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛r❡ ✷✲❞❡❣❡♥❡r❛t❡ = ⇒ ✸✲❝♦❧♦✉r❛❜❧❡✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ◗✉❡st✐♦♥ ❲❤❛t ✐s ❛♥ ♦♣t✐♠❛❧ ❜♦✉♥❞ ♦❢ ♦❞❞✲❣✐rt❤ ✷ ❦ + ✶ ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶❄ Pr♦♣♦s✐t✐♦♥ ❆ ❣r❛♣❤ ✐s ❑ ✹ ✲♠✐♥♦r ❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ♣❛rt✐❛❧ ✷✲tr❡❡✳ ❘❡♠❛r❦ ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛r❡ ✷✲❞❡❣❡♥❡r❛t❡ = ⇒ ✸✲❝♦❧♦✉r❛❜❧❡✳ ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s✿ ❛❧♠♦st ❡q✉✐✈❛❧❡♥t t♦ s❡r✐❡s✲♣❛r❛❧❧❡❧ ❣r❛♣❤s✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✸ ✴ ✸✽

❈✐r❝✉❧❛r ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✹ ✴ ✸✽

❈✐r❝✉❧❛r ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❉❡✜♥✐t✐♦♥ ✲ ♣ q ✲❝♦❧♦✉r✐♥❣ ♦❢ ● ▼❛♣♣✐♥❣ ❝ : ❱ ( ● ) → { ✶ ,..., ♣ } s✳t✳ ①② ∈ ❊ ( ● ) ⇒ q ≤ | ❝ ( ① ) − ❝ ( ② ) | ≤ ♣ − q ✳ ❈✐r❝✉❧❛r ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r✿ χ ❝ ( ● ) = ✐♥❢ { ♣ q | ● ✐s ♣ q ✲❝♦❧♦✉r❛❜❧❡ } ❘❡♠❛r❦ • ❊q✉✐✈❛❧❡♥t❧②✱ ❤♦♠♦♠♦r♣❤✐s♠ t♦ ❝✐r❝✉❧❛r ❝❧✐q✉❡ ❑ ( ♣ / q ) • ✷ ❦ + ✶ ✲❝♦❧♦✉r✐♥❣ ⇐ ⇒ ❤♦♠♦♠♦r♣❤✐s♠ t♦ ❈ ✷ ❦ + ✶ ❦ • ❘❡✜♥❡♠❡♥t ♦❢ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r✿ χ ( ● ) − ✶ < χ ❝ ( ● ) ≤ χ ( ● ) ❚❤❡♦r❡♠ ✭❍❡❧❧ ✫ ❩❤✉✱ ✷✵✵✵ ✰ P❛♥ ✫ ❩❤✉✱ ✷✵✵✷✮ ■❢ ● ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❛♥❞ tr✐❛♥❣❧❡✲❢r❡❡✱ χ ❝ ( ● ) ≤ ✽ ✸ ✳ ■❢ ♠♦r❡♦✈❡r ● ❤❛s ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✼✱ χ ❝ ( ● ) ≤ ✺ ✷ ✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✺ ✴ ✸✽

●❡♥❡r❛❧ ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✻ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦r♦❧❧❛r② ❊✈❡r② ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ( ✷ ❦ + ✶ ) ✲❣r❛♣❤ ✐s ( ✷ ❦ + ✶ ) ✲❡❞❣❡✲❝♦❧♦✉r❛❜❧❡✳ − → ❆ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t ❛❧r❡❛❞② ♣r♦✈❡❞ ❜② ❙❡②♠♦✉r ✭✶✾✾✵✮ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ❑♥❡s❡r ❣r❛♣❤ ✭✏♦❞❞ ❣r❛♣❤✑✮ ❑♥ ( ✷ ❦ + ✶ , ❦ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❑ ✹ ✲♠✐♥♦r✲ ❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ✶ , ✷ ❑♥❡s❡r ❣r❛♣❤ ❑♥ ( ❛ , ❜ ) ✿ ✈❡rt✐❝❡s ❛r❡ ❜ ✲s✉❜s❡ts ♦❢ { ✶ ,..., ❛ } ✸ , ✺ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❞✐s❥♦✐♥t✳ ✹ , ✺ ✸ , ✹ ✷ , ✺ ✶ , ✸ ❊①❛♠♣❧❡✿ ❑♥ ( ✺ , ✷ ) = P❡t❡rs❡♥ ❣r❛♣❤✳ ✷ , ✹ ✶ , ✹ ✷ , ✸ ✶ , ✺ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ❑♥❡s❡r ❣r❛♣❤ ✭✏♦❞❞ ❣r❛♣❤✑✮ ❑♥ ( ✷ ❦ + ✶ , ❦ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❑ ✹ ✲♠✐♥♦r✲ ❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ✶ , ✷ ❑♥❡s❡r ❣r❛♣❤ ❑♥ ( ❛ , ❜ ) ✿ ✈❡rt✐❝❡s ❛r❡ ❜ ✲s✉❜s❡ts ♦❢ { ✶ ,..., ❛ } ✸ , ✺ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❞✐s❥♦✐♥t✳ ✹ , ✺ ✸ , ✹ ✷ , ✺ ✶ , ✸ ❊①❛♠♣❧❡✿ ❑♥ ( ✺ , ✷ ) = P❡t❡rs❡♥ ❣r❛♣❤✳ ✷ , ✹ ✶ , ✹ ✷ , ✸ ✶ , ✺ ❈♦r♦❧❧❛r② ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶ ❤❛✈❡ ❢r❛❝t✐♦♥❛❧ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ❛t ♠♦st ✷ + ✶ ❦ ✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ✷ ❦ × ✷ ❦ ♣r♦❥❡❝t✐✈❡ t♦r♦✐❞❛❧ ❣r✐❞ P❚● ✷ ❦ , ✷ ❦ ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❑ ✹ ✲♠✐♥♦r✲ ❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ✷ ❦ × ✷ ❦ ♣r♦❥❡❝t✐✈❡ t♦r♦✐❞❛❧ ❣r✐❞ P❚● ✷ ❦ , ✷ ❦ ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❑ ✹ ✲♠✐♥♦r✲ ❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ P❚● ✹ , ✹ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ✷ ❦ × ✷ ❦ ♣r♦❥❡❝t✐✈❡ t♦r♦✐❞❛❧ ❣r✐❞ P❚● ✷ ❦ , ✷ ❦ ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❑ ✹ ✲♠✐♥♦r✲ ❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ P❚● ✹ , ✹ = P❈ ( ✹ ) ✭❈❧❡❜s❝❤ ❣r❛♣❤✮ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ✷ ❦ × ✷ ❦ ♣r♦❥❡❝t✐✈❡ t♦r♦✐❞❛❧ ❣r✐❞ P❚● ✷ ❦ , ✷ ❦ ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❑ ✹ ✲♠✐♥♦r✲ ❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ P❚● ✹ , ✹ = P❈ ( ✹ ) ✭❈❧❡❜s❝❤ ❣r❛♣❤✮ ❦ = ✸✿ P❚● ✻ , ✻ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ✷ ❦ × ✷ ❦ ♣r♦❥❡❝t✐✈❡ t♦r♦✐❞❛❧ ❣r✐❞ P❚● ✷ ❦ , ✷ ❦ ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❑ ✹ ✲♠✐♥♦r✲ ❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ P❚● ✹ , ✹ = P❈ ( ✹ ) ✭❈❧❡❜s❝❤ ❣r❛♣❤✮ ❦ = ✸✿ P❚● ✻ , ✻ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ●röt③s❝❤ ❣r❛♣❤ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ●röt③s❝❤ ❣r❛♣❤ ❦ = ✸✿ ▼ ✷ ( ❈ ✼ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ●röt③s❝❤ ❣r❛♣❤ ❦ = ✸✿ ▼ ✷ ( ❈ ✼ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ●röt③s❝❤ ❣r❛♣❤ ❦ = ✸✿ ▼ ✷ ( ❈ ✼ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ●röt③s❝❤ ❣r❛♣❤ ❦ = ✸✿ ▼ ✷ ( ❈ ✼ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ●röt③s❝❤ ❣r❛♣❤ ❦ = ✸✿ ▼ ✷ ( ❈ ✼ ) ❦ = ✹✿ ▼ ✸ ( ❈ ✾ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ●röt③s❝❤ ❣r❛♣❤ ❦ = ✸✿ ▼ ✷ ( ❈ ✼ ) ❦ = ✹✿ ▼ ✸ ( ❈ ✾ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ●röt③s❝❤ ❣r❛♣❤ ❦ = ✸✿ ▼ ✷ ( ❈ ✼ ) ❦ = ✹✿ ▼ ✸ ( ❈ ✾ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ●röt③s❝❤ ❣r❛♣❤ ❦ = ✸✿ ▼ ✷ ( ❈ ✼ ) ❦ = ✹✿ ▼ ✸ ( ❈ ✾ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽

❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡s ❈♦♥❥❡❝t✉r❡ ✭◆❛s❡r❛sr✱ ✷✵✵✼✮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❝✉❜❡ P❈ ( ✷ ❦ ) ❜♦✉♥❞s t❤❡ ❝❧❛ss ♦❢ ♣❧❛♥❛r ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❈♦♥❥❡❝t✉r❡ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ◆❛s❡r❛sr✱ ✷✵✶✼✮ ❚❤❡ ●❡♥❡r❛❧✐③❡❞ ▼②❝✐❡❧s❦✐❛♥ ♦❢ ❧❡✈❡❧ ❦ − ✶ ♦❢ ❈ ✷ ❦ + ✶ ▼ ❦ − ✶ ( ❈ ✷ ❦ + ✶ ) ⊂ P❈ ( ✷ ❦ ) ❜♦✉♥❞s ❢♦r ❑ ✹ ✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ♦❢ ♦❞❞✲❣✐rt❤ ❛t ❧❡❛st ✷ ❦ + ✶✳ ❦ = ✷✿ ▼ ✶ ( ❈ ✺ ) ●röt③s❝❤ ❣r❛♣❤ ❦ = ✸✿ ▼ ✷ ( ❈ ✼ ) ❦ = ✹✿ ▼ ✸ ( ❈ ✾ ) ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ❍♦♠♦♠♦r♣❤✐s♠ ❜♦✉♥❞s ❢♦r ❑ ✹✲♠✐♥♦r✲❢r❡❡ ❣r❛♣❤s ✷✼ ✴ ✸✽