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❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ✭❯♥✐✈❡rs✐t② ♦❢ Pr✐♠♦rs❦❛✱ ❙❧♦✈❡♥✐❛✮ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❊❞✇❛r❞ ❉♦❜s♦♥✱ ▼❛rt✐♥ ▼✐❧❛♥✐↔ ❛♥❞ ●❛❜r✐❡❧ ❱❡rr❡t✳ ✶✻✳✵✻✳✷✵✶✺✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s

❖✈❡r✈✐❡✇ ❈■❙ ❣r❛♣❤s ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s ✇✐t❤ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ♦❢ s✐③❡ ✷ ♦r ✸ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s ♦❢ ✈❛❧❡♥❝② ❛t ♠♦st ✼ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s

❚❤❡ ✐♥❝❧✉s✐♦♥ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ st❛❜❧❡ s❡ts ♦❢ ❛r❡ ❝❛❧❧❡❞ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ♠❛①✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② ❛ st❛❜❧❡ s❡t✮ ♦❢ ✐s ❝❛❧❧❡❞ t❤❡ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② st❛❜✐❧✐t② ✮ ♥✉♠❜❡r ❛♥❞ ❞❡♥♦t❡❞ ✭r❡s♣❡❝t✐✈❡❧② ✮✳ ❉❡✜♥✐t✐♦♥ ❆ ❈■❙ ❣r❛♣❤ ✐s ❛ ❣r❛♣❤ ✐♥ ✇❤✐❝❤ ❡✈❡r② ♠❛①✐♠❛❧ st❛❜❧❡ s❡t ❛♥❞ ❡✈❡r② ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐♥t❡rs❡❝t ✭❈■❙ st❛♥❞s ❢♦r ✏❈❧✐q✉❡s ■♥t❡rs❡❝t ❙t❛❜❧❡ s❡ts✑✮✳ ❈■❙ ❣r❛♣❤s ▲❡t Γ ❞❡♥♦t❡ ❛ ✜♥✐t❡ s✐♠♣❧❡ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❆ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧②✱ ❛ st❛❜❧❡ s❡t ✮ ♦❢ Γ ✐s ❛ s❡t ♦❢ ♣❛✐r✇✐s❡ ❛❞❥❛❝❡♥t ✭r❡s♣❡❝t✐✈❡❧②✱ ♥♦♥✲❛❞❥❛❝❡♥t✮ ✈❡rt✐❝❡s✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥ ❆ ❈■❙ ❣r❛♣❤ ✐s ❛ ❣r❛♣❤ ✐♥ ✇❤✐❝❤ ❡✈❡r② ♠❛①✐♠❛❧ st❛❜❧❡ s❡t ❛♥❞ ❡✈❡r② ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐♥t❡rs❡❝t ✭❈■❙ st❛♥❞s ❢♦r ✏❈❧✐q✉❡s ■♥t❡rs❡❝t ❙t❛❜❧❡ s❡ts✑✮✳ ❈■❙ ❣r❛♣❤s ▲❡t Γ ❞❡♥♦t❡ ❛ ✜♥✐t❡ s✐♠♣❧❡ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❆ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧②✱ ❛ st❛❜❧❡ s❡t ✮ ♦❢ Γ ✐s ❛ s❡t ♦❢ ♣❛✐r✇✐s❡ ❛❞❥❛❝❡♥t ✭r❡s♣❡❝t✐✈❡❧②✱ ♥♦♥✲❛❞❥❛❝❡♥t✮ ✈❡rt✐❝❡s✳ ❚❤❡ ✐♥❝❧✉s✐♦♥ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ st❛❜❧❡ s❡ts ♦❢ Γ ❛r❡ ❝❛❧❧❡❞ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ♠❛①✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② ❛ st❛❜❧❡ s❡t✮ ♦❢ Γ ✐s ❝❛❧❧❡❞ t❤❡ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② st❛❜✐❧✐t② ✮ ♥✉♠❜❡r ❛♥❞ ❞❡♥♦t❡❞ ω (Γ) ✭r❡s♣❡❝t✐✈❡❧② α (Γ) ✮✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s

❈■❙ ❣r❛♣❤s ▲❡t Γ ❞❡♥♦t❡ ❛ ✜♥✐t❡ s✐♠♣❧❡ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❆ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧②✱ ❛ st❛❜❧❡ s❡t ✮ ♦❢ Γ ✐s ❛ s❡t ♦❢ ♣❛✐r✇✐s❡ ❛❞❥❛❝❡♥t ✭r❡s♣❡❝t✐✈❡❧②✱ ♥♦♥✲❛❞❥❛❝❡♥t✮ ✈❡rt✐❝❡s✳ ❚❤❡ ✐♥❝❧✉s✐♦♥ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ st❛❜❧❡ s❡ts ♦❢ Γ ❛r❡ ❝❛❧❧❡❞ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ♠❛①✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② ❛ st❛❜❧❡ s❡t✮ ♦❢ Γ ✐s ❝❛❧❧❡❞ t❤❡ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② st❛❜✐❧✐t② ✮ ♥✉♠❜❡r ❛♥❞ ❞❡♥♦t❡❞ ω (Γ) ✭r❡s♣❡❝t✐✈❡❧② α (Γ) ✮✳ ❉❡✜♥✐t✐♦♥ ❆ ❈■❙ ❣r❛♣❤ ✐s ❛ ❣r❛♣❤ ✐♥ ✇❤✐❝❤ ❡✈❡r② ♠❛①✐♠❛❧ st❛❜❧❡ s❡t ❛♥❞ ❡✈❡r② ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐♥t❡rs❡❝t ✭❈■❙ st❛♥❞s ❢♦r ✏❈❧✐q✉❡s ■♥t❡rs❡❝t ❙t❛❜❧❡ s❡ts✑✮✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s

❊①❛♠♣❧❡ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s

❆ ❞✐s❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡❛❝❤ ♦❢ ✐ts ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐s ❈■❙✳ ❋♦r ❡✈❡r② t✇♦ ❣r❛♣❤s ✶ ❛♥❞ ✷ ✱ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ✷ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✶ ❛♥❞ ✷ ❛r❡ ❈■❙✳ ✶ ❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❣r❛♣❤s ✶ ❛♥❞ ✷ ✐s t❤❡ ❣r❛♣❤ ✶ ✷ ✇✐t❤ ✈❡rt❡① s❡t ✷ ✱ ✇❤❡r❡ t✇♦ ✈❡rt✐❝❡s ❛♥❞ ✶ ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r ✶ ♦r ❛♥❞ ✷ ✳ ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈■❙ ❣r❛♣❤s ❆ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❝♦♠♣❧❡♠❡♥t ✐s ❈■❙✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s

❋♦r ❡✈❡r② t✇♦ ❣r❛♣❤s ✶ ❛♥❞ ✷ ✱ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ✷ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✶ ❛♥❞ ✷ ❛r❡ ❈■❙✳ ✶ ❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❣r❛♣❤s ✶ ❛♥❞ ✷ ✐s t❤❡ ❣r❛♣❤ ✶ ✷ ✇✐t❤ ✈❡rt❡① s❡t ✷ ✱ ✇❤❡r❡ t✇♦ ✈❡rt✐❝❡s ❛♥❞ ✶ ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r ✶ ♦r ❛♥❞ ✷ ✳ ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈■❙ ❣r❛♣❤s ❆ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❝♦♠♣❧❡♠❡♥t ✐s ❈■❙✳ ❆ ❞✐s❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡❛❝❤ ♦❢ ✐ts ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐s ❈■❙✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s

❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈■❙ ❣r❛♣❤s ❆ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❝♦♠♣❧❡♠❡♥t ✐s ❈■❙✳ ❆ ❞✐s❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡❛❝❤ ♦❢ ✐ts ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐s ❈■❙✳ ❋♦r ❡✈❡r② t✇♦ ❣r❛♣❤s Γ ✶ ❛♥❞ Γ ✷ ✱ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t Γ ✶ [Γ ✷ ] ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Γ ✶ ❛♥❞ Γ ✷ ❛r❡ ❈■❙✳ ❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❣r❛♣❤s Γ ✶ ❛♥❞ Γ ✷ ✐s t❤❡ ❣r❛♣❤ Γ ✶ [Γ ✷ ] ✇✐t❤ ✈❡rt❡① s❡t V (Γ ✶ ) × V (Γ ✷ ) ✱ ✇❤❡r❡ t✇♦ ✈❡rt✐❝❡s ( u , x ) ❛♥❞ ( v , y ) ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r { u , v } ∈ E (Γ ✶ ) ♦r u = v ❛♥❞ { x , y } ∈ E (Γ ✷ ) ✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s

✭❈♦✮✲✇❡❧❧✲❝♦✈❡r❡❞ ❣r❛♣❤s ❆ ❣r❛♣❤ Γ ✐s ✇❡❧❧✲❝♦✈❡r❡❞ ✐❢ ❛❧❧ ✐ts ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts ❛r❡ ♦❢ t❤❡ s❛♠❡ s✐③❡✳ ❆ ❣r❛♣❤ Γ ✐s ❝♦✲✇❡❧❧✲❝♦✈❡r❡❞ ✐❢ ❛❧❧ ✐ts ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛r❡ ♦❢ t❤❡ s❛♠❡ s✐③❡✳ ✭❊q✉✐✈❛❧❡♥t❧②✿ ✐❢ ✐ts ❝♦♠♣❧❡♠❡♥t ✐s ✇❡❧❧✲❝♦✈❡r❡❞✳✮ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s