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  1. ❲♦r❦s❤♦♣ ♦♥ ●❡♦♠❡tr② ▼❡t❤♦❞s ✐♥ P❤②s✐❝s ❇✐❛➟♦✇✐❡➺❛ ✷✵✶✾ ❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ G ✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s ●r③❡❣♦r③ ❏❛❦✐♠♦✇✐❝③ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❯♥✐✈❡rs✐t② ✐♥ ❇✐❛➟②st♦❦ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆♥❡t❛ ❙❧✐➺❡✇s❦❛ ❛♥❞ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ✒❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ G ✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s✧✱ ❛r❳✐✈✿✶✼✶✷✳✵✽✹✷✵ t♦ ❛♣♣❡❛r ✐♥ ❘❡✈✐❡✇s ✐♥ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s ❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

  2. ❚❤❡ ♣❤❛s❡ s♣❛❝❡ ♦❢ ❛ t②♣✐❝❛❧ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ✐s t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ T ∗ P ♦❢ ✐ts ❝♦♥✜❣✉r❛t✐♦♥s s♣❛❝❡ P ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ st❛♥❞❛r❞ s②♠♣❧❡❝t✐❝ ❢♦r♠✳ ❯s✉❛❧❧② ♦♥❡ ❝♦♥s✐❞❡rs t❤❡ ❝❛s❡ ✇❤❡♥ ❍❛♠✐❧t♦♥✐❛♥ ♦❢ t❤✐s s②st❡♠ ✐s ✐♥✈❛r✐❛♥t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦t❛♥❣❡♥t ❧✐❢t ♦❢ ❛♥ ❛❝t✐♦♥ ♦❢ s♦♠❡ ❣r♦✉♣ G ♦♥ P ✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s s♦♠❡t✐♠❡s r❡❛s♦♥❛❜❧❡ t♦ r❡♣❧❛❝❡ t❤❡ st❛♥❞❛r❞ s②♠♣❧❡❝t✐❝ ❢♦r♠ ✇✐t❤ ❛♥♦t❤❡r G ✲✐♥✈❛r✐❛♥t s②♠♣❧❡❝t✐❝ ❢♦r♠ ♦♥ T ∗ P ✇❤✐❝❤ r❡t❛✐♥s ❝❡rt❛✐♥ ♣r♦♣❡rt✐❡s✳ ❆ss✉♠✐♥❣ t❤❛t t❤❡ ❛❝t✐♦♥ ♦❢ G ♦♥ P ✐s ❢r❡❡ ❛♥❞ t❤❡ q✉♦t✐❡♥t s♣❛❝❡ P/G ✐s ❛ ♠❛♥✐❢♦❧❞ M ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r P ❛s t❤❡ t♦t❛❧ s♣❛❝❡ ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ G ✲❜✉♥❞❧❡ P ( M, G ) ✳ ❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

  3. ❚❤❡ ❣r♦✉♣ ✭❛♥❞ ✐ts s✉❜❣r♦✉♣s✮ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❛❝t✐♦♥ ♦❢ ♦♥ ✳ ❚❤❡ ❣r♦✉♣ ❛❝ts ♦♥ t❤❡ ❜♦t❤ s♣❛❝❡s✿ ❚❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ✲❜✉♥❞❧❡ ✳ ❚❤❡ s♣❛❝❡ ♦❢ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠s ♦♥ t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ ✱ ✇❤✐❝❤ ❛♥♥✐❤✐❧❛t❡ t❤❡ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜r❡s ♦❢ ✳ ▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ P ( M, G ) ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳ ❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

  4. ❚❤❡ ❣r♦✉♣ ❛❝ts ♦♥ t❤❡ ❜♦t❤ s♣❛❝❡s✿ ❚❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ✲❜✉♥❞❧❡ ✳ ❚❤❡ s♣❛❝❡ ♦❢ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠s ♦♥ t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ ✱ ✇❤✐❝❤ ❛♥♥✐❤✐❧❛t❡ t❤❡ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜r❡s ♦❢ ✳ ▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ P ( M, G ) ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳ • ❚❤❡ ❣r♦✉♣ Aut TG TP ✭❛♥❞ ✐ts s✉❜❣r♦✉♣s✮ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TP ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ P ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❛❝t✐♦♥ ♦❢ TG ♦♥ TP ✳ ❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

  5. ❚❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ✲❜✉♥❞❧❡ ✳ ❚❤❡ s♣❛❝❡ ♦❢ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠s ♦♥ t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ ✱ ✇❤✐❝❤ ❛♥♥✐❤✐❧❛t❡ t❤❡ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜r❡s ♦❢ ✳ ▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ P ( M, G ) ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳ • ❚❤❡ ❣r♦✉♣ Aut TG TP ✭❛♥❞ ✐ts s✉❜❣r♦✉♣s✮ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TP ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ P ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❛❝t✐♦♥ ♦❢ TG ♦♥ TP ✳ ❚❤❡ ❣r♦✉♣ Aut TG TP ❛❝ts ♦♥ t❤❡ ❜♦t❤ s♣❛❝❡s✿ ❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

  6. ❚❤❡ s♣❛❝❡ ♦❢ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠s ♦♥ t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ ✱ ✇❤✐❝❤ ❛♥♥✐❤✐❧❛t❡ t❤❡ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜r❡s ♦❢ ✳ ▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ P ( M, G ) ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳ • ❚❤❡ ❣r♦✉♣ Aut TG TP ✭❛♥❞ ✐ts s✉❜❣r♦✉♣s✮ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TP ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ P ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❛❝t✐♦♥ ♦❢ TG ♦♥ TP ✳ ❚❤❡ ❣r♦✉♣ Aut TG TP ❛❝ts ♦♥ t❤❡ ❜♦t❤ s♣❛❝❡s✿ • ❚❤❡ s♣❛❝❡ ConnP ( M, G ) ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ G ✲❜✉♥❞❧❡ P ( M, G ) ✳ ❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

  7. ▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ P ( M, G ) ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳ • ❚❤❡ ❣r♦✉♣ Aut TG TP ✭❛♥❞ ✐ts s✉❜❣r♦✉♣s✮ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TP ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ P ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❛❝t✐♦♥ ♦❢ TG ♦♥ TP ✳ ❚❤❡ ❣r♦✉♣ Aut TG TP ❛❝ts ♦♥ t❤❡ ❜♦t❤ s♣❛❝❡s✿ • ❚❤❡ s♣❛❝❡ ConnP ( M, G ) ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ G ✲❜✉♥❞❧❡ P ( M, G ) ✳ • ❚❤❡ s♣❛❝❡ CanT ∗ P ♦❢ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠s γ ♦♥ t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ T ∗ P ✱ ✇❤✐❝❤ ❛♥♥✐❤✐❧❛t❡ t❤❡ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜r❡s ♦❢ T ∗ P ✳ ❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

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