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❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s ◆✐❝♦❧❛s ❇♦✉sq✉❡t✱ ❆♥t♦✐♥❡ ❉❛✐❧❧②✱ ❱❛❧❡♥t✐♥ ●❧❡❞❡❧ ❛♥❞ ▼❛r❝ ❍❡✐♥r✐❝❤ ❯♥✐✈❡rs✐té ❞❡ ▲②♦♥ ❈●❚❈ ✲ ❏❛♥✉❛r② t❤❡ ✷✼t❤✱ ✷✵✶✼ Games And Graphs ✶✴✷✼

❆r❝✲❑❛②❧❡s ✷✴✷✼

❆r❝✲❑❛②❧❡s ❆r❝✲❑❛②❧❡s ✿ ❙❝❤❛❡❢❡r✱ ✶✾✼✽ ❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) ✳ ❼ ❆t ❡❛❝❤ t✉r♥ t❤❡ ❝✉rr❡♥t ♣❧❛②❡r ❝❤♦♦s❡s ❛♥ ❡❞❣❡✳ ❼ ■ts ❡♥❞♣♦✐♥ts ❛r❡ ❞❡❧❡t❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ✸✴✷✼

❆r❝✲❑❛②❧❡s ❆r❝✲❑❛②❧❡s ✿ ❙❝❤❛❡❢❡r✱ ✶✾✼✽ ❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) ✳ ❼ ❆t ❡❛❝❤ t✉r♥ t❤❡ ❝✉rr❡♥t ♣❧❛②❡r ❝❤♦♦s❡s ❛♥ ❡❞❣❡✳ ❼ ■ts ❡♥❞♣♦✐♥ts ❛r❡ ❞❡❧❡t❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ✸✴✷✼

❆r❝✲❑❛②❧❡s ❆r❝✲❑❛②❧❡s ✿ ❙❝❤❛❡❢❡r✱ ✶✾✼✽ ❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) ✳ ❼ ❆t ❡❛❝❤ t✉r♥ t❤❡ ❝✉rr❡♥t ♣❧❛②❡r ❝❤♦♦s❡s ❛♥ ❡❞❣❡✳ ❼ ■ts ❡♥❞♣♦✐♥ts ❛r❡ ❞❡❧❡t❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ✸✴✷✼

❆r❝✲❑❛②❧❡s ❆r❝✲❑❛②❧❡s ✿ ❙❝❤❛❡❢❡r✱ ✶✾✼✽ ❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) ✳ ❼ ❆t ❡❛❝❤ t✉r♥ t❤❡ ❝✉rr❡♥t ♣❧❛②❡r ❝❤♦♦s❡s ❛♥ ❡❞❣❡✳ ❼ ■ts ❡♥❞♣♦✐♥ts ❛r❡ ❞❡❧❡t❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ✸✴✷✼

❆r❝✲❑❛②❧❡s ❼ ❙♦❧✈❡❞ ♦♥ ♣❛t❤s ✭●✉② ❛♥❞ ❙♠✐t❤✱ ✶✾✺✻✮ ❼ ❙♦❧✈❡❞ ♦♥ ❡q✉✐♠❛t❝❤❛❜❧❡ ❣r❛♣❤s✱ ❝②❝❧❡s✱ ✇❤❡❡❧s ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ st❛r ❣r❛♣❤s ✇✐t❤ t❤r❡❡ ❜r❛♥❝❤❡s ✭❍✉❣❣❛♥ ❛♥❞ ❙t❡✈❡♥s✱ ✷✵✶✻✮ ❼ ❋P❚ ✭▲❛♠♣✐s ❛♥❞ ▼✐ts♦✉✱ ✷✵✶✹✮ ✹✴✷✼

❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s ✺✴✷✼

✶ ✶ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s ❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) ✇✐t❤ ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ω : V → N ✳ ❼ ❆t ❡❛❝❤ st❡♣ t❤❡ ❝✉rr❡♥t ♣❧❛②❡r ❝❤♦♦s❡ ❛♥ ❡❞❣❡✳ ❼ ❚❤❡ ✇❡✐❣❤t ♦❢ ❜♦t❤ ❡♥❞♣♦✐♥ts ✐s ❞❡❝r❡❛s❡❞ ❜② ✶✳ m n ❼ ❱❡rt✐❝❡s ✇✐t❤ ✇❡✐❣❤t ③❡r♦ ❛r❡ r❡♠♦✈❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ✻✴✷✼

✶ ✶ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s ❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) ✇✐t❤ ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ω : V → N ✳ ❼ ❆t ❡❛❝❤ st❡♣ t❤❡ ❝✉rr❡♥t ♣❧❛②❡r ❝❤♦♦s❡ ❛♥ ❡❞❣❡✳ ❼ ❚❤❡ ✇❡✐❣❤t ♦❢ ❜♦t❤ ❡♥❞♣♦✐♥ts ✐s ❞❡❝r❡❛s❡❞ ❜② ✶✳ m n ❼ ❱❡rt✐❝❡s ✇✐t❤ ✇❡✐❣❤t ③❡r♦ ❛r❡ r❡♠♦✈❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ✻✴✷✼

❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s ❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛ ❣r❛♣❤ G = ( V , E ) ✇✐t❤ ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ω : V → N ✳ ❼ ❆t ❡❛❝❤ st❡♣ t❤❡ ❝✉rr❡♥t ♣❧❛②❡r ❝❤♦♦s❡ ❛♥ ❡❞❣❡✳ ❼ ❚❤❡ ✇❡✐❣❤t ♦❢ ❜♦t❤ ❡♥❞♣♦✐♥ts ✐s ❞❡❝r❡❛s❡❞ ❜② ✶✳ m − ✶ n − ✶ ❼ ❱❡rt✐❝❡s ✇✐t❤ ✇❡✐❣❤t ③❡r♦ ❛r❡ r❡♠♦✈❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ✻✴✷✼

✷ ✸ ✶ ✸ ✸ ✶ ✸ ✸ ✵ ❊①❛♠♣❧❡ ✸ ✹ ✶ ✼✴✷✼

✸ ✸ ✶ ✸ ✸ ✵ ❊①❛♠♣❧❡ ✸ ✹ ✶ ✷ ✸ ✶ ✼✴✷✼

✸ ✸ ✵ ❊①❛♠♣❧❡ ✸ ✹ ✶ ✷ ✸ ✶ ✸ ✸ ✶ ✼✴✷✼

❊①❛♠♣❧❡ ✸ ✹ ✶ ✷ ✸ ✶ ✸ ✸ ✶ ✸ ✸ ✵ ✼✴✷✼

✵ ❊①❛♠♣❧❡ ✸ ✹ ✶ ✷ ✸ ✶ ✸ ✸ ✶ ✸ ✸ ✼✴✷✼

◆♦♥✲❛tt❛❝❦✐♥❣ r♦♦❦s ✐♥ ❛ ❤♦❧❡❞ ❝❤❡ss❜♦❛r❞ ✽✴✷✼

❚❤❡ ❣❛♠❡ ✻ 0Z0Z0Z0Z ✺ Z0Z0S0Z0 ✹ 0Z0Z0Z0Z ✸ S0Z0Z0Z0 ✷ 0Z0Z0Z0Z ✶ Z0Z0Z0Z0 ❛ ❜ ❝ ❞ ❡ ❢ ❣ ❤ ■♥s♣✐r❡❞ ❜② t❤❡ ♥♦♥✲❛tt❛❝❦✐♥❣ q✉❡❡♥s ❣❛♠❡✭◆♦♦♥ ❛♥❞ ❱❛♥ ❇r✉♠♠❡❧❡♥✱ ✷✵✵✻✮ ✾✴✷✼

❚❤❡ ❣❛♠❡ ✶✵ 0Z0Z0Z0Z0Z ✾ Z0Z0Z0Z0Z0 ✽ 0Z0Z0Z0Z0Z ✼ Z0Z0Z0Z0Z0 ✻ 0S0Z0Z0S0Z ✺ Z0Z0Z0Z0Z0 ✹ 0Z0Z0Z0Z0Z ✸ Z0Z0ZRZ0Z0 ✷ 0Z0Z0Z0Z0Z ✶ Z0Z0Z0Z0Z0 ❛ ❜ ❝ ❞ ❡ ❢ ❣ ❤ ✐ ❥ ✶✵✴✷✼

✷ ✷ ❚❤❡ ❣❛♠❡ ✶✵ 0Z0Z0Z0Z0Z ✷ ✷ ✾ Z0Z0Z0Z0Z0 ✸ ✽ 0Z0Z0Z0Z0Z ✹ ✼ Z0Z0Z0Z0Z0 ✻ 0S0Z0Z0S0Z ✷ ✷ ✺ Z0Z0Z0Z0Z0 ✹ 0Z0Z0Z0Z0Z ✸ ✸ Z0Z0ZRZ0Z0 ✷ 0Z0Z0Z0Z0Z ✸ ✶ Z0Z0Z0Z0Z0 ❛ ❜ ❝ ❞ ❡ ❢ ❣ ❤ ✐ ❥ ✶✵✴✷✼

✸ ✸ ❚❤❡ ❣❛♠❡ ✶✵ 0Z0Z0Z0Z0Z ✷ ✷ ✾ Z0Z0Z0Z0Z0 ✸ ✽ 0Z0Z0Z0Z0Z ✹ ✼ Z0Z0Z0Z0Z0 ✻ 0S0Z0Z0S0Z ✷ ✷ ✺ Z0Z0Z0Z0Z0 ✹ 0Z0Z0Z0Z0Z ✷ ✸ Z0Z0ZRZ0Z0 ✷ 0Z0ZRZ0Z0Z ✷ ✶ Z0Z0Z0Z0Z0 ❛ ❜ ❝ ❞ ❡ ❢ ❣ ❤ ✐ ❥ ✶✵✴✷✼

✷ ✷ ✶ ✷ ✶ ✷ ✶ ✶ ✶ ✷ ✶ ❘❡❞✉❝t✐♦♥ t♦ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s n ✶ w ✷ n ✷ m ✷ h ✶ h ✷ m ✶ w ✶ ✶✶✴✷✼

✷ ✷ ✶ ✷ ✶ ✷ ✶ ✶ ✶ ✷ ✶ ❘❡❞✉❝t✐♦♥ t♦ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s n ✶ w ✷ n ✷ m ✷ h ✶ h ✷ m ✶ w ✶ ✶✶✴✷✼

✶ ✷ ✶ ❘❡❞✉❝t✐♦♥ t♦ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s n ✶ w ✷ n ✷ w ✷ m ✷ m ✷ n ✶ n ✷ h ✶ h ✷ h ✶ h ✷ m ✶ m ✶ w ✶ w ✶ ✶✶✴✷✼

✷ ✷ ✷ ✷ ✶ ✷ ✶ ❘❡❞✉❝t✐♦♥ t♦ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s n ✶ n ✷ h ✶ h ✷ n ✶ n ✷ h ✶ h ✷ m ✶ m ✶ w ✶ w ✶ ✶✶✴✷✼

❋✐rst r❡s✉❧ts ✶✷✴✷✼

✷ ✷ ✶ ✷ ✷ ✷ ✶ ✷ ❆❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥ ▲❡♠♠❛ ■❢ ❛❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥✱ t❤❡♥ t❤❡ ♦✉t❝♦♠❡ ✐s P ✷ m ✷ n ✶✸✴✷✼

✷ ✷ ✷ ✷ ✷ ✷ ❆❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥ ▲❡♠♠❛ ■❢ ❛❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥✱ t❤❡♥ t❤❡ ♦✉t❝♦♠❡ ✐s P ✷ m − ✶ ✷ n − ✶ ✶✸✴✷✼

✷ ✷ ✶ ✷ ✷ ✶ ❆❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥ ▲❡♠♠❛ ■❢ ❛❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥✱ t❤❡♥ t❤❡ ♦✉t❝♦♠❡ ✐s P ✷ m − ✷ ✷ n − ✷ ✶✸✴✷✼

✶ ✶ ❘❡❞✉❝t✐♦♥ ❧❡♠♠❛s ❚✇✐♥ ✈❡rt✐❝❡s ❧❡♠♠❛ ■❢ t✇♦ ✈❡rt✐❝❡s ❤❛✈❡ t❤❡ s❛♠❡ ♥❡✐❣❤❜♦rs ❛♥❞ ❡✐t❤❡r ❤❛✈❡ ❜♦t❤ ❧♦♦♣s ♦r ♥♦♥❡ ♦❢ t❤❡♠ ❤❛✈❡ ❧♦♦♣s ✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❞✉❝t✐♦♥ ❤♦❧❞s ✿ m n m + n ✶✹✴✷✼

❘❡❞✉❝t✐♦♥ ❧❡♠♠❛s ❚✇✐♥ ✈❡rt✐❝❡s ❧❡♠♠❛ ■❢ t✇♦ ✈❡rt✐❝❡s ❤❛✈❡ t❤❡ s❛♠❡ ♥❡✐❣❤❜♦rs ❛♥❞ ❡✐t❤❡r ❤❛✈❡ ❜♦t❤ ❧♦♦♣s ♦r ♥♦♥❡ ♦❢ t❤❡♠ ❤❛✈❡ ❧♦♦♣s ✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❞✉❝t✐♦♥ ❤♦❧❞s ✿ m − ✶ n m + n − ✶ ✶✹✴✷✼