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  1. ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ✭❯♥✐✈❡rs✐t② ♦❢ Pr✐♠♦rs❦❛✱ ❙❧♦✈❡♥✐❛✮ ❏♦✐♥t ✇♦r❦ ✇✐t❤ Pr✐♠♦➸ P♦t♦↔♥✐❦ ❛♥❞ ●❛❜r✐❡❧ ❱❡rr❡t ✶✳✻✳✷✵✶✽ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  2. ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  3. ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  4. ❉r❛❣❛♥ ▼❛r✉➨✐↔ ❛♥❞ ❤✐s P❤❉ st✉❞❡♥ts ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  5. ❆✉t♦♠♦r♣❤✐s♠ ♦❢ ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ♦❢ ♣r❡s❡r✈✐♥❣ t❤❡ ❛❞❥❛❝❡♥❝② r❡❧❛t✐♦♥✳ ❆✉t ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❛♥❞ ✐s ❝❛❧❧❡❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ ✳ ❆ ❣r❛♣❤ ✐s s❛✐❞ t♦ ❜❡ ✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ ✲❡❞❣❡✲tr❛♥s✐t✐✈❡ ♦r ✲❛r❝✲tr❛♥s✐t✐✈❡ ✐❢ ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ ✱ ♦r ❆✉t ✱ r❡s♣❡❝t✐✈❡❧②✳ ❇❛s✐❝ ❞❡✜♥✐t✐♦♥s ▲❡t Γ ❜❡ ❛ s✐♠♣❧❡✱ ✜♥✐t❡ ❛♥❞ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❲✐t❤ V (Γ) ✱ E (Γ) ❛♥❞ A (Γ) ✇❡ ❞❡♥♦t❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ✈❡rt❡①✲s❡t✱ t❤❡ ❡❞❣❡✲s❡t ❛♥❞ t❤❡ ❛r❝✲s❡t ♦❢ Γ ✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  6. ❆ ❣r❛♣❤ ✐s s❛✐❞ t♦ ❜❡ ✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ ✲❡❞❣❡✲tr❛♥s✐t✐✈❡ ♦r ✲❛r❝✲tr❛♥s✐t✐✈❡ ✐❢ ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ ✱ ♦r ❆✉t ✱ r❡s♣❡❝t✐✈❡❧②✳ ❇❛s✐❝ ❞❡✜♥✐t✐♦♥s ▲❡t Γ ❜❡ ❛ s✐♠♣❧❡✱ ✜♥✐t❡ ❛♥❞ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❲✐t❤ V (Γ) ✱ E (Γ) ❛♥❞ A (Γ) ✇❡ ❞❡♥♦t❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ✈❡rt❡①✲s❡t✱ t❤❡ ❡❞❣❡✲s❡t ❛♥❞ t❤❡ ❛r❝✲s❡t ♦❢ Γ ✳ ❆✉t♦♠♦r♣❤✐s♠ ♦❢ Γ ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ♦❢ V (Γ) ♣r❡s❡r✈✐♥❣ t❤❡ ❛❞❥❛❝❡♥❝② r❡❧❛t✐♦♥✳ ❆✉t (Γ) ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ Γ ❛♥❞ ✐s ❝❛❧❧❡❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ Γ ✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  7. ❇❛s✐❝ ❞❡✜♥✐t✐♦♥s ▲❡t Γ ❜❡ ❛ s✐♠♣❧❡✱ ✜♥✐t❡ ❛♥❞ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❲✐t❤ V (Γ) ✱ E (Γ) ❛♥❞ A (Γ) ✇❡ ❞❡♥♦t❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ✈❡rt❡①✲s❡t✱ t❤❡ ❡❞❣❡✲s❡t ❛♥❞ t❤❡ ❛r❝✲s❡t ♦❢ Γ ✳ ❆✉t♦♠♦r♣❤✐s♠ ♦❢ Γ ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ♦❢ V (Γ) ♣r❡s❡r✈✐♥❣ t❤❡ ❛❞❥❛❝❡♥❝② r❡❧❛t✐♦♥✳ ❆✉t (Γ) ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ Γ ❛♥❞ ✐s ❝❛❧❧❡❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ Γ ✳ ❆ ❣r❛♣❤ Γ ✐s s❛✐❞ t♦ ❜❡ G ✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ G ✲❡❞❣❡✲tr❛♥s✐t✐✈❡ ♦r G ✲❛r❝✲tr❛♥s✐t✐✈❡ ✐❢ G ≤ ❆✉t (Γ) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ V (Γ) ✱ E (Γ) ♦r A (Γ) ✱ r❡s♣❡❝t✐✈❡❧②✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  8. ▲❛r❣❡r ♦❜❥❡❝ts ❤❛✈❡ t❤❡ ♣♦t❡♥t✐❛❧ t♦ ❛❞♠✐t ♠✉❝❤ ❧❛r❣❡r ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s✳ ■❢ ✐s ✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ t❤❡♥ ✳ ▼❡❛s✉r✐♥❣ ❤♦✇ s②♠♠❡tr✐❝ ❣r❛♣❤ ✐s ❆ ♥❛t✉r❛❧ ❛♣♣r♦❛❝❤ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  9. ■❢ ✐s ✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ t❤❡♥ ✳ ▼❡❛s✉r✐♥❣ ❤♦✇ s②♠♠❡tr✐❝ ❣r❛♣❤ ✐s ❆ ♥❛t✉r❛❧ ❛♣♣r♦❛❝❤ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳ ▲❛r❣❡r ♦❜❥❡❝ts ❤❛✈❡ t❤❡ ♣♦t❡♥t✐❛❧ t♦ ❛❞♠✐t ♠✉❝❤ ❧❛r❣❡r ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  10. ▼❡❛s✉r✐♥❣ ❤♦✇ s②♠♠❡tr✐❝ ❣r❛♣❤ ✐s ❆ ♥❛t✉r❛❧ ❛♣♣r♦❛❝❤ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳ ▲❛r❣❡r ♦❜❥❡❝ts ❤❛✈❡ t❤❡ ♣♦t❡♥t✐❛❧ t♦ ❛❞♠✐t ♠✉❝❤ ❧❛r❣❡r ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s✳ ■❢ Γ ✐s G ✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ t❤❡♥ | G | = | V (Γ) | · | G v | ✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  11. ❈✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ❆ ❣r❛♣❤ ✐s s❛✐❞ t♦ ❜❡ ❝✉❜✐❝ ✐❢ ❡❛❝❤ ✈❡rt❡① ✐s ✐♥❝✐❞❡♥t ✇✐t❤ t❤r❡❡ ❡❞❣❡s✳ ❚❤❡♦r❡♠ ✭❚✉tt❡✱ ✶✾✹✻✮ ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ❝✉❜✐❝ ✭✸✲r❡❣✉❧❛r✮ G ✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ❛♥❞ ❧❡t v ∈ V (Γ) ✳ ❚❤❡♥ | G v | = ✸ ❛♥❞ G ❛❝ts r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ❛r❝s ♦❢ Γ ❀ | G v | = ✻ ❛♥❞ G ❛❝ts r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ✷✲❛r❝s ♦❢ Γ ❀ | G v | = ✶✷ ❛♥❞ G ❛❝ts r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ✸✲❛r❝s ♦❢ Γ ❀ | G v | = ✷✹ ❛♥❞ G ❛❝ts r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ✹✲❛r❝s ♦❢ Γ ❀ | G v | = ✹✽ ❛♥❞ G ❛❝ts r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ✺✲❛r❝s ♦❢ Γ ✳ ❋♦r ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡r s ✱ ❛♥ s ✲❛r❝ ✐♥ ❛ ❣r❛♣❤ Γ ✐s ❛♥ ( s + ✶ ) ✲t✉♣❧❡ ♦❢ ✈❡rt✐❝❡s v ✵ , v ✶ , . . . , v s ✱ s✉❝❤ t❤❛t v i − ✶ ✐s ❛❞❥❛❝❡♥t ✇✐t❤ v i ✭❢♦r ✶ ≤ i ≤ s ✮ ❛♥❞ v i − ✶ � = v i + ✶ ✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

  12. ▲❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❣r❛♣❤s ❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✐s t❤❡ ❣r❛♣❤ G [ H ] ✇✐t❤ ✈❡rt❡① s❡t V ( G ) × V ( H ) ✱ ✇❤❡r❡ t✇♦ ✈❡rt✐❝❡s ( x ✶ , y ✶ ) ❛♥❞ ( x ✷ , y ✷ ) ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r { x ✶ , x ✷ } ∈ E ( G ) ♦r x ✶ = x ✷ ❛♥❞ { y ✶ , y ✷ } ∈ E ( H ) ✳ ❊①❛♠♣❧❡ C ✻ [ K ✷ ] ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s

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