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❖♣t✐♠✐s✐♥❣ r❡❛❝❤❛❜✐❧✐t② ❜② r❡♦r❞❡r✐♥❣ ❑✐tt② ▼❡❡❦s ❙❝❤♦♦❧ ♦❢ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡✱ ❯♥✐✈❡rs✐t② ♦❢ ●❧❛s❣♦✇ ❆❧❣♦r✐t❤♠✐❝ ❆s♣❡❝ts ♦❢ ❚❡♠♣♦r❛❧ ●r❛♣❤s ■■✱ P❛tr❛s✱ ●r❡❡❝❡ ✽t❤ ❏✉❧② ✷✵✶✾ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❏❡ss✐❝❛ ❊♥r✐❣❤t ✭❯♥✐✈❡rs✐t② ♦❢ ❊❞✐♥❜✉r❣❤✮ ❛♥❞ ❋✐♦♥❛ ❙❦❡r♠❛♥ ✭▼❛s❛r②❦ ❯♥✐✈❡rs✐t②✮

❘❡❛❝❤❛❜✐❧✐t② s❡ts ✐♥ t❡♠♣♦r❛❧ ❣r❛♣❤s 1 1 2 4,1 2 3 3,5 1,2,3 4,3 2

❖✉r ❣♦❛❧ ✐s t♦ ♠✐♥✐♠✐s❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❧❛r❣❡st r❡❛❝❤❛❜✐❧✐t② s❡t✳ ❆ss✐❣♥✐♥❣ t✐♠❡s t♦ r❡❞✉❝❡ r❡❛❝❤❛❜✐❧✐t② ◮ ❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡✱ ✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ❛❝t✐✈❡ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✳

❆ss✐❣♥✐♥❣ t✐♠❡s t♦ r❡❞✉❝❡ r❡❛❝❤❛❜✐❧✐t② ◮ ❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡✱ ✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ❛❝t✐✈❡ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✳ ◮ ❖✉r ❣♦❛❧ ✐s t♦ ♠✐♥✐♠✐s❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❧❛r❣❡st r❡❛❝❤❛❜✐❧✐t② s❡t✳

❆ss✐❣♥✐♥❣ t✐♠❡s t♦ r❡❞✉❝❡ r❡❛❝❤❛❜✐❧✐t② ◮ ❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡✱ ✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ❛❝t✐✈❡ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✳ ◮ ❖✉r ❣♦❛❧ ✐s t♦ ♠✐♥✐♠✐s❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❧❛r❣❡st r❡❛❝❤❛❜✐❧✐t② s❡t✳

❙■◆●▲❊❚❖◆ ▼■◆✲▼❆❳ ❘❊❆❈❍❆❇■▲■❚❨ ❚❊▼P❖❘❆▲ ❖❘❉❊❘■◆● ■♥♣✉t✿ ❆ ❣r❛♣❤ ✱ ❛♥❞ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛ ❜✐❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t ♠❛①✐♠✉♠ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② ♦❢ ✐s ❛t ♠♦st ❄ ❖✉r ❜❛s✐❝ ♣r♦❜❧❡♠ ❚❤❡ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❛ ✈❡rt❡① v ✐♥ ❛ t❡♠♣♦r❛❧ ❣r❛♣❤ ( G , T ) ✐s t❤❡ s❡t ♦❢ ✈❡rt✐❝❡s u s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛ str✐❝t t❡♠♣♦r❛❧ ♣❛t❤ ❢r♦♠ v t♦ u ❀ ✇❡ ✐♥❝❧✉❞❡ v ✐♥ t❤✐s s❡t✳ ❚❤❡ ♠❛①✐♠✉♠ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② ♦❢ ❛ t❡♠♣♦r❛❧ ❣r❛♣❤ ✐s t❤❡ ♠❛①✐♠✉♠ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❛♥② ✈❡rt❡① v ✐♥ t❤❡ ❣r❛♣❤✳

❖✉r ❜❛s✐❝ ♣r♦❜❧❡♠ ❚❤❡ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❛ ✈❡rt❡① v ✐♥ ❛ t❡♠♣♦r❛❧ ❣r❛♣❤ ( G , T ) ✐s t❤❡ s❡t ♦❢ ✈❡rt✐❝❡s u s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛ str✐❝t t❡♠♣♦r❛❧ ♣❛t❤ ❢r♦♠ v t♦ u ❀ ✇❡ ✐♥❝❧✉❞❡ v ✐♥ t❤✐s s❡t✳ ❚❤❡ ♠❛①✐♠✉♠ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② ♦❢ ❛ t❡♠♣♦r❛❧ ❣r❛♣❤ ✐s t❤❡ ♠❛①✐♠✉♠ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❛♥② ✈❡rt❡① v ✐♥ t❤❡ ❣r❛♣❤✳ ❙■◆●▲❊❚❖◆ ▼■◆✲▼❆❳ ❘❊❆❈❍❆❇■▲■❚❨ ❚❊▼P❖❘❆▲ ❖❘❉❊❘■◆● ■♥♣✉t✿ ❆ ❣r❛♣❤ G = ( V , E ) ✱ ❛♥❞ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r k ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛ ❜✐❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ T : E → [ | E | ] s✉❝❤ t❤❛t ♠❛①✐♠✉♠ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② ♦❢ ( G , T ) ✐s ❛t ♠♦st k ❄

❲❡ ❣✐✈❡ ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ ▼✐♥✐♠✉♠ ❇✐s❡❝t✐♦♥ ❢♦r ❈✉✲ ❜✐❝ ●r❛♣❤s✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ❤❛r❞ ❚❤❡♦r❡♠ ❙■◆●▲❊❚❖◆ ▼■◆✲▼❆❳ ❘❊❆❈❍❆❇■▲■❚❨ ❚❊▼P❖❘❆▲ ❖❘❉❊❘■◆● ✐s ◆P✲❝♦♠♣❧❡t❡✳

❚❤❡ ♣r♦❜❧❡♠ ✐s ❤❛r❞ h-2r-21 ... (r,h-r-10)-decoration (r,h-r-10)-decoration b a v 1 v 1 w 1 h-2r-21 ... (r,h-r-10)-decoration (r,h-r-10)-decoration ❚❤❡♦r❡♠ a b v 2 v 2 w 2 ❙■◆●▲❊❚❖◆ ▼■◆✲▼❆❳ ... ❘❊❆❈❍❆❇■▲■❚❨ h-2r-21 ... ❚❊▼P❖❘❆▲ ❖❘❉❊❘■◆● (r,h-r-10)-decoration (r,h-r-10)-decoration a b v i v i ✐s ◆P✲❝♦♠♣❧❡t❡✳ w i v i -v j h-gadget ... v i -v j h-gadget if and only if if and only if v i v j is an edge v i v j is an edge h-2r-21 ... (r,h-r-10)-decoration (r,h-r-10)-decoration ❲❡ ❣✐✈❡ ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ a b v j v j w j ▼✐♥✐♠✉♠ ❇✐s❡❝t✐♦♥ ❢♦r ❈✉✲ ... ❜✐❝ ●r❛♣❤s✳ h-2r-21 ... (r,h-r-10)-decoration (r,h-r-10)-decoration a b v n v n w n x a x b

❚❤❡♦r❡♠ ❚❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ❢♦r ❛ ❉❆● ✐s ❡①❛❝t❧② t❤❡ ♠❛①✐♠✉♠ ♦✉t✲❞❡❣r❡❡ ♣❧✉s ♦♥❡✳ ●❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ◮ ❍♦✇❡✈❡r ✇❡ ♦r❞❡r t❤❡ ❡❞❣❡s✱ t❤❡ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❡✈❡r② ✈❡rt❡① ❝♦♥t❛✐♥s ❛❧❧ ♦❢ ✐ts ✭♦✉t✲✮♥❡✐❣❤❜♦✉rs✳

●❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ◮ ❍♦✇❡✈❡r ✇❡ ♦r❞❡r t❤❡ ❡❞❣❡s✱ t❤❡ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❡✈❡r② ✈❡rt❡① ❝♦♥t❛✐♥s ❛❧❧ ♦❢ ✐ts ✭♦✉t✲✮♥❡✐❣❤❜♦✉rs✳ ❚❤❡♦r❡♠ ❚❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ❢♦r ❛ ❉❆● ✐s ❡①❛❝t❧② t❤❡ ♠❛①✐♠✉♠ ♦✉t✲❞❡❣r❡❡ ♣❧✉s ♦♥❡✳

●❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ◮ ❍♦✇❡✈❡r ✇❡ ♦r❞❡r t❤❡ ❡❞❣❡s✱ t❤❡r❡ ✐s ❛ r❡❛❝❤❛❜✐❧✐t② s❡t ✇❤✐❝❤ ❝♦♥t❛✐♥s ❛❧❧ ♥❡✐❣❤❜♦✉rs ♦❢ t✇♦ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s✳

●❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ◮ ❍♦✇❡✈❡r ✇❡ ♦r❞❡r t❤❡ ❡❞❣❡s✱ t❤❡r❡ ✐s ❛ r❡❛❝❤❛❜✐❧✐t② s❡t ✇❤✐❝❤ ❝♦♥t❛✐♥s ❛❧❧ ♥❡✐❣❤❜♦✉rs ♦❢ t✇♦ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s✳ ❚❤❡♦r❡♠ ❋♦r ❛ ♣❛t❤ ♦♥ ❛t ❧❡❛st ✜✈❡ ✈❡rt✐❝❡s✱ t❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ✐s ❡①❛❝t❧② ❢♦✉r✳

●❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ◮ ❚❤❡r❡ ✐s ❛♥ ♦r❞❡r ♦❢ ❡❞❣❡s t❤❛t ♠✐♥✐♠✐s❡s t❤❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t②✱ s♦ t❤❛t ❛❧❧ ✏❧❡❛❢✑ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡ ❜❡❢♦r❡ ❛❧❧ ♦t❤❡rs✳ ▼♦r❡♦✈❡r✱ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ♦❢ ❧❡❛❢ ❡❞❣❡s ❞♦❡s ♥♦t ♠❛tt❡r✳

●❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ◮ ❚❤❡r❡ ✐s ❛♥ ♦r❞❡r ♦❢ ❡❞❣❡s t❤❛t ♠✐♥✐♠✐s❡s t❤❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t②✱ s♦ t❤❛t ❛❧❧ ✏❧❡❛❢✑ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡ ❜❡❢♦r❡ ❛❧❧ ♦t❤❡rs✳ ▼♦r❡♦✈❡r✱ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ♦❢ ❧❡❛❢ ❡❞❣❡s ❞♦❡s ♥♦t ♠❛tt❡r✳ ❚❤❡♦r❡♠ ❚❤❡r❡ ✐s ❛♥ ❋P❚ ❛❧❣♦r✐t❤♠ t♦ ❝♦♠♣✉t❡ t❤❡ ♦♣t✐♠✉♠ ♦r❞❡r✐♥❣ ♦❢ ❡❞❣❡s✱ ♣❛r❛♠❡t❡r✐s❡❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ♦❢ ❞❡❣r❡❡ ❛t ❧❡❛st t✇♦✳