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❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍♦♥❣✇❡✐ ●❛♦ 1 ▲❡♦♥ P❡tr♦s②❛♥ 2 ❆rt❡♠ ❙❡❞❛❦♦✈ 2 1 ❈♦❧❧❡❣❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ◗✐♥❣❞❛♦ ❯♥✐✈❡rs✐t② 2 ❉❡♣❛rt♠❡♥t ♦❢ ●❛♠❡ ❚❤❡♦r② ❛♥❞ ❙t❛t✐st✐❝❛❧ ❉❡❝✐s✐♦♥s ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ❏✉♥❡ ✸✱ ✷✵✶✹ ❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✴✸✵

❖✉t❧✐♥❡ ✶ ❚❤❡ ♠♦❞❡❧ ✷ ❈♦♦♣❡r❛t✐♦♥ ✐♥ t✇♦✲st❛❣❡ ♥❡t✇♦r❦ ❣❛♠❡ ✸ ❙tr♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t s♦❧✉t✐♦♥ ✹ ◆✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡ ❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✴✸✵

❚❤❡ ▼♦❞❡❧ N = { 1 , . . . , n } ✿ ❛ ✜♥✐t❡ s❡t ♦❢ ♣❧❛②❡rs ✇❤♦ ❝❛♥ ✐♥t❡r❛❝t ✇✐t❤ ❡❛❝❤ ♦t❤❡r✳ g ✿ ❛ ✜♥✐t❡ s❡t ♦❢ ♣❛✐rs ( i , j ) ∈ N × N ✱ ♦r ❛ ♥❡t✇♦r❦✳ ■❢ ( i , j ) ∈ g ✱ ✇❡ s❛② t❤❛t t❤❡r❡ ✐s ❛ ❧✐♥❦ ❝♦♥♥❡❝t✐♥❣ ♣❧❛②❡rs i ❛♥❞ j ✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♦❢ t❤❡ ♣❧❛②❡rs✳ ■♥ ♦✉r s❡tt✐♥❣ ✇❡ s✉♣♣♦s❡ t❤❛t ❛❧❧ ❧✐♥❦s ❛r❡ ✉♥❞✐r❡❝t❡❞✱ ✐✳❡✳ ( i , j ) = ( j , i ) ✳ ❋♦r t❤❡ s✐♠♣❧✐❝✐t② ❞❡♥♦t❡ ( i , j ) ❛s ij ✳ ❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✸✴✸✵

❈♦♥s✐❞❡r ❛ t✇♦✲st❛❣❡ ♣r♦❜❧❡♠✿ ❆t t❤❡ ✜rst st❛❣❡ ❡❛❝❤ ♣❧❛②❡r ❝❤♦♦s❡s ❤✐s ♣❛rt♥❡rs✖♦t❤❡r ♣❧❛②❡rs ✇✐t❤ ✇❤♦♠ ❤❡ ✇❛♥ts t♦ ❢♦r♠ ❧✐♥❦s✳ ❆❢t❡r ❝❤♦♦s✐♥❣ ♣❛rt♥❡rs ❛♥❞ ❡st❛❜❧✐s❤✐♥❣ ❧✐♥❦s✱ ♣❧❛②❡rs✱ t❤❡r❡❜②✱ ❢♦r♠ ❛ ♥❡t✇♦r❦✳ ❆t t❤❡ s❡❝♦♥❞ st❛❣❡ ❤❛✈✐♥❣ t❤❡ ♥❡t✇♦r❦ ❢♦r♠❡❞✱ ❡❛❝❤ ♣❧❛②❡r ❝❤♦♦s❡s ❛ ❝♦♥tr♦❧ ✐♥✢✉❡♥❝✐♥❣ ❤✐s ♣❛②♦✛✳ ❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✹✴✸✵

❋✐rst ❙t❛❣❡✿ ◆❡t✇♦r❦ ❋♦r♠❛t✐♦♥ M i ⊆ N \ { i } ✿ t❤❡ s❡t ♦❢ ♣❧❛②❡rs ✇❤♦♠ ♣❧❛②❡r i ∈ N ❝❛♥ ♦✛❡r ❛ ♠✉t✉❛❧ ❧✐♥❦✳ a i ∈ { 0 , . . . , n − 1 } ✿ t❤❡ ♠❛①✐♠❛❧ ♥✉♠❜❡r ♦❢ ❧✐♥❦s ✇❤✐❝❤ ♣❧❛②❡r i ❝❛♥ ♦✛❡r✳ ❇❡❤❛✈✐♦r ♦❢ ♣❧❛②❡r i ∈ N ❛t t❤❡ ✜rst st❛❣❡ ✐s ❛ ♣r♦✜❧❡ g i = ( g i 1 , . . . , g in ) ✇❤✐❝❤ ❝♦♠♣♦♥❡♥ts ❛r❡ ❞❡✜♥❡❞ ❛s✿ � 1 , ✐❢ ♣❧❛②❡r i ♦✛❡rs ❛ ❧✐♥❦ t♦ j ∈ M i , g ij = ✭✶✮ 0 , ♦t❤❡r✇✐s❡ , s✉❜❥❡❝t t♦ t❤❡ ❝♦♥str❛✐♥✿ � ✭✷✮ g ij � a i , j ∈ N g ii = 0 , i ∈ N . G i = { g i : (1) − (2) ❛r❡ tr✉❡ } ✱ i ∈ N ✳ ❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✺✴✸✵

P❧❛②❡rs ❝❤♦♦s❡ t❤❡✐r ❜❡❤❛✈✐♦rs ❛t t❤❡ ✜rst st❛❣❡ s✐♠✉❧t❛♥❡♦✉s❧② ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ ❡❛❝❤ ♦t❤❡r✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ♣❧❛②❡r i ∈ N ❝❤♦♦s❡s g i ∈ G i ✱ ❛♥❞ ❛s ❛ r❡s✉❧t t❤❡ ❜❡❤❛✈✐♦r ♣r♦✜❧❡ ( g 1 , . . . , g n ) ✐s ❢♦r♠❡❞✳ ❍❛✈✐♥❣ t❤❡ ❜❡❤❛✈✐♦r ♣r♦✜❧❡ ( g 1 , . . . , g n ) ❢♦r♠❡❞✱ ❛♥ ✉♥❞✐r❡❝t❡❞ ❧✐♥❦ ij = ji ✐s ❡st❛❜❧✐s❤❡❞ ✐♥ ♥❡t✇♦r❦ g ✐❢ ❛♥❞ ♦♥❧② ✐❢ g ij = g ji = 1 , ✐✳❡✳ g ❝♦♥s✐sts ♦❢ ♠✉t✉❛❧ ❧✐♥❦s ✇❤✐❝❤ ✇❡r❡ ♦✛❡r❡❞ ♦♥❧② ❜② ❜♦t❤ ♣❧❛②❡rs✳ ❊①❛♠♣❧❡ ▲❡t N = { 1 , 2 , 3 , 4 } ❛♥❞ ♣❧❛②❡rs ❝❤♦♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❜❡❤❛✈✐♦rs ❛t t❤❡ ✜rst st❛❣❡✿ g 1 = (0 , 1 , 1 , 1) ✱ g 2 = (1 , 0 , 1 , 0) ✱ g 3 = (1 , 1 , 0 , 0) ✱ g 4 = (0 , 0 , 1 , 0) ✳ ❚❤❡ r❡s✉❧t✐♥❣ ♥❡t✇♦r❦ g ❝♦♥t❛✐♥s t❤r❡❡ ❧✐♥❦s { 12 , 13 , 23 } ✳ ❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✻✴✸✵

❙❡❝♦♥❞ ❙t❛❣❡✿ ❈♦♥tr♦❧s N i ( g ) = { j ∈ N \ { i } : ij ∈ g } ✿ ♥❡✐❣❤❜♦rs ♦❢ ♣❧❛②❡r i ✐♥ ♥❡t✇♦r❦ g ✳ ▲❡t d i ( g ) = ( d i 1 ( g ) , . . . , d in ( g )) ❜❡ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿  1 , ✐❢ i ❞♦❡s ♥♦t ❜r❡❛❦ t❤❡ ❧✐♥❦ ❢♦r♠❡❞   ❛t t❤❡ ✜rst st❛❣❡  d ij ( g ) = ✭✸✮ ✇✐t❤ ♣❧❛②❡r j ∈ N i ( g ) ✐♥ ♥❡t✇♦r❦ g ,   0 , ♦t❤❡r✇✐s❡ .  D i ( g ) = { d i ( g ) : (3) ✐s tr✉❡ } ✳ Pr♦✜❧❡ ( d 1 ( g ) , . . . , d n ( g )) ❝❤❛♥❣❡s ♥❡t✇♦r❦ g ❛♥❞ ❢♦r♠s ❛ ♥❡✇ ♥❡t✇♦r❦✱ ❞❡♥♦t❡❞ ❜② g d ✳ ■♥ ♥❡t✇♦r❦ g d ❛❧❧ ❧✐♥❦s ij s✉❝❤ t❤❛t ❡✐t❤❡r d ij ( g ) = 0 ♦r d ji ( g ) = 0 ❛r❡ r❡♠♦✈❡❞✳ ❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✼✴✸✵

❆t t❤❡ s❡❝♦♥❞ st❛❣❡ ♣❧❛②❡r i ∈ N ❝❤♦♦s❡s ❝♦♥tr♦❧ u i ❢r♦♠ ❛ ✜♥✐t❡ s❡t U i ✳ ❇❡❤❛✈✐♦r ♦❢ ♣❧❛②❡r i ∈ N ❛t t❤❡ s❡❝♦♥❞ st❛❣❡ ✐s ❛ ♣❛✐r ( d i ( g ) , u i ) ✿ ✐t ❞❡✜♥❡s✱ ♦♥ t❤❡ ♦♥❡ ❤❛♥❞✱ ❧✐♥❦s t♦ ❜❡ r❡♠♦✈❡❞ d i ( g ) ✱ ❛♥❞✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❝♦♥tr♦❧ u i ✳ P❛②♦✛ ❢✉♥❝t✐♦♥ K i ♦❢ ♣❧❛②❡r i ✿ ✐t r✉❧❡s✱ ❞❡♣❡♥❞s ♦♥ ♣❧❛②❡r✬s ❜❡❤❛✈✐♦r ❛t t❤❡ s❡❝♦♥❞ st❛❣❡ ❛s ✇❡❧❧ ❛s ❜❡❤❛✈✐♦r ♦❢ ❤✐s ♥❡✐❣❤❜♦rs ✐♥ ♥❡t✇♦r❦ g d ✳ K i ( u i , u N i ( g d ) ) ✱ i ∈ N ✱ ✐s ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ s❡t U i × � j ∈ N i ( g d ) U j ❛♥❞ s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✿ ✭P✮✿ ❢♦r ❛♥② t✇♦ ♥❡t✇♦r❦s g ❛♥❞ g ′ ❛♥❞ ♣❧❛②❡r i ✐❢ | N i ( g ) | � | N i ( g ′ ) | ✱ t❤❡ ✐♥❡q✉❛❧✐t② K i ( u i , u N i ( g ) ) � K i ( u i , u N i ( g ′ ) ) ❤♦❧❞s ❢♦r ❛❧❧ ( u i , u N i ( g ) ) ∈ U i × � j ∈ N i ( g ) U j ❛♥❞ ( u i , u N i ( g ′ ) ) ∈ U i × � j ∈ N i ( g ′ ) U j ✳ ❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✽✴✸✵

❈♦♦♣❡r❛t✐♦♥ ✐♥ ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡ ❲❡ st✉❞② t❤❡ ❝♦♦♣❡r❛t✐✈❡ ❝❛s❡ ❛♥❞ ❛♥s✇❡r t❤r❡❡ ♠❛✐♥ q✉❡st✐♦♥s✿ ❲❤❛t ✐s ❛ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ✐♥ t❤❡ ❣❛♠❡❄ ❈❛♥ ✐t ❜❡ r❡❛❧✐③❡❞ ✐♥ t❤❡ ❣❛♠❡❄ ■s ✐t str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t❄ ❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✾✴✸✵