Projeto e Análise de Algoritmos Grafos

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❘✉r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦

Pr♦❥❡t♦ ❡ ❆♥á❧✐s❡ ❞❡ ❆❧❣♦r✐t♠♦s

●r❛❢♦s

Pr♦❢✳ ❚✐❛❣♦ ❇✉❛rq✉❡ ❆ss✉♥çã♦ ❞❡ ❈❛r✈❛❧❤♦

❯♥✐❞❛❞❡ ❆❝❛❞ê♠✐❝❛ ❞❡ ●❛r❛♥❤✉♥s ✕ ❯❋❘P❊ ❇❛❝❤❛r❡❧❛❞♦ ❡♠ ❈✐ê♥❝✐❛s ❞❛ ❈♦♠♣✉t❛çã♦

✶✹ ❞❡ ❞❡③❡♠❜r♦ ❞❡ ✷✵✶✽

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❘♦t❡✐r♦

✶ ●r❛❢♦

✷ Pr♦♣r✐❡❞❛❞❡s

✸ ❚✐♣♦s

✹ ➪r✈♦r❡s

✺ ●r❛❢♦s ♣❧❛♥❛r❡s

✻ ❈♦❧♦r❛çã♦ ❞❡ ●r❛❢♦s

✼ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ●r❛❢♦s

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❆❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s

• _{▲♦❣íst✐❝❛✿}

• _{❈❛♠✐♥❤♦s ♥♦s ♠❛♣❛s✳}

• _{●r❛♥❞❡s r❡str✐çõ❡s ♥♦ ❡♥✈✐♦ ❞❡ ♠❡r❝❛❞♦r✐❛s✳} • _{❘❡❞❡s s♦❝✐❛✐s✳}

• _{❇❛♥❝♦s ❞❡ ❞❛❞♦s✳}

• _{▼♦❞❡❧❛❣❡♠ ❞♦s r❡❧❛❝✐♦♥❛♠❡♥t♦s ♣❛r❛ ✈✐s✉❛❧✐③❛çã♦ ❞❡ ❝♦♥s✉❧t❛s✳} • _{❇❛♥❝♦s ♦r✐❡♥t❛❞♦s ❛ ❣r❛❢♦s✳}

▲✐♥❦ ❞❛ r❡♣♦rt❛❣❡♠

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●r❛❢♦

• _{❯♠ ❣r❛❢♦}G(V, E) é ❝♦♠♣♦st♦ ♣♦r ❞♦✐s ❡❧❡♠❡♥t♦s✿

✭✐✮ ❯♠ ❝♦♥❥✉♥t♦V ₌V₍G₎ ❞❡ ✈ért✐❝❡s ✭♦✉ ♥ós✮ ❞❡G✳

✭✐✐✮ ❯♠ ❝♦♥❥✉♥t♦E ₌E₍G₎ ❞❡ ♣❛r❡s ♥ã♦ ♦r❞❡♥❛❞♦s ❞❡ ✈ért✐❝❡s ❞✐st✐♥t♦s✱ ❝❤❛♠❛❞♦s ❛r❡st❛s ❞❡G✳

• _{❖s ✈ért✐❝❡s} u ❡v sã♦ ❞✐t♦s ❛❞❥❛❝❡♥t❡s s❡ e∈E✱ e={u, v}.

• u ❡v sã♦ ❞✐t♦s ❡①tr❡♠♦s ❞❡ e✳ • e❝♦♥❡❝t❛u ❛v✳

• eé ✐♥❝✐❞❡♥t❡u ❛v✳

• _{❯♠ ♠✉❧t✐❣r❛❢♦ ❞✐❢❡r❡ ❞❡ ✉♠ ❣r❛❢♦ s✐♠♣❧❡s ♣♦r ♣♦ss✉✐r✿} • _{❆r❡st❛s ♠ú❧t✐♣❧❛s ❝♦♥❡❝t❛♥❞♦ ♦ ♠❡s♠♦ ♣❛r ❞❡ ✈ért✐❝❡s✳} • _{▲❛ç♦s ❝♦♥❡❝t❛♥❞♦ ✉♠ ✈ért✐❝❡ ❛ ❡❧❡ ♠❡s♠♦✳}

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●r❛✉ ❞❡ ✉♠ ✈ért✐❝❡

• deg(v) é ♦ ❣r❛✉ ❞❡ ✉♠ ✈ért✐❝❡v✱ ✐st♦ é✱ ♦ ♥ú♠❡r♦ ❞❡

❡①tr❡♠✐❞❛❞❡s ❞❡ ❛r❡st❛s q✉❡ ✐♥❝✐❞❡ ♥❛q✉❡❧❡ ♥ó✳

• _{❈❛❞❛ ❛r❡st❛ é ❝♦♥t❛❞❛ ❞✉❛s ✈❡③❡s ✭❝❛❞❛ ❡①tr❡♠✐❞❛❞❡ ❞❡ ✉♠❛} ❛r❡st❛✮✳

• _{❚❡♦r❡♠❛ s♦❜r❡ ♦s ❣r❛✉s ❞♦s ✈ért✐❝❡s ❡ ♥ú♠❡r♦s ❞❡} ❛r❡st❛s✿ ❛ s♦♠❛ ❞♦s ❣r❛✉s ❞❡ ✉♠ ❣r❛❢♦ é ✐❣✉❛❧ ❛ ❞✉❛s ✈❡③❡s ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✳

• _{❊①❡♠♣❧♦ ✭❛✮✿} deg(A) = 2✱ deg(B) = 3✱deg(C) = 3✱ deg(D) = 2.

• _{❊①❡♠♣❧♦ ✭❜✮✿} deg(A) = 2✱deg(B) = 3✱ deg(C) = 3✱

deg(D) = 4. ✭❖ ❧❛ç♦ ❝♦♥t❛ ❞✉❛s ✈❡③❡s✱ ❝❛❞❛ ❡①tr❡♠✐❞❛❞❡ ❝♦♥t❛

✉♠❛ ✈❡③✮✳

• _{❱ért✐❝❡ ♣❛r é ❛q✉❡❧❡ q✉❡ t❡♠ ❣r❛✉ ♣❛r✳} • _{❱ért✐❝❡ ✐s♦❧❛❞♦ é ❛q✉❡❧❡ q✉❡ t❡♠ ❣r❛✉ ③❡r♦✳}

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●r❛❢♦s ✜♥✐t♦s ❡ ●r❛❢♦ tr✐✈✐❛❧

• _{●r❛❢♦ ✜♥✐t♦}

• _{❯♠ ♠✉❧t✐❣r❛❢♦ ✜♥✐t♦ t❡♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈ért✐❝❡s ❡ ❛r❡st❛s✳} • _{❯♠ ❣r❛❢♦ s✐♠♣❧❡s ❝♦♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈ért✐❝❡s t❛♠❜é♠}

t❡rá ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❛r❡st❛s✱ ♣♦rt❛♥t♦ é ✜♥✐t♦✳ • _{❯♠ ❣r❛❢♦ tr✐✈✐❛❧ t❡♠ ✉♠ ú♥✐❝♦ ✈ért✐❝❡ ❡ ♥❡♥❤✉♠❛ ❛r❡st❛✳}

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❙✉❜❣r❛❢♦

H(V′_{, E}′₎é ✉♠ s✉❜❣r❛❢♦ ❞❡ _G₍_{V, E}₎ s❡_V′ _⊆_V ❡_E′ _⊆_E✳

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●r❛❢♦s ■s♦♠♦r❢♦s

G(V, E) ❡ G∗₍_V∗_{, E}∗₎sã♦ ❞✐t♦s ✐s♦♠♦r❢♦s s❡✿ • _{❡①✐st❡}f :V →V∗✱

• _s❡{u, v} ∈E s❡ ❡ s♦♠❡♥t❡ s❡{f(u), f(v)} ∈E∗✳

❙ã♦ ✐s♦♠♦r❢♦s✿ ❆ ❡ ❘❀ ❋ ❡ ❚❀ ❑ ❡ ❳❀ ▼✱ ❙✱ ❱ ❡ ❩✳

❯♠❛ ❢✉♥çã♦f:A→Bé ❜✐❥❡t♦r❛ q✉❛♥❞♦fé ✐♥❥❡t♦r❛ ❡ s♦❜r❡❥❡t♦r❛✳ ◆❛ ❢✉♥çã♦ ✐♥❥❡t♦r❛✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❛

♣r✐♠❡✐r❛ tê♠ ❝♦♠♦ ✐♠❛❣❡♠ ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❞❛ ♦✉tr❛✳ ❏á ♥❛ ❢✉♥çã♦ s♦❜r❡❥❡t♦r❛✱ t♦❞♦ ❡❧❡♠❡♥t♦ ❞♦

❝♦♥tr❛❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦ é ✐♠❛❣❡♠ ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❞♦♠í♥✐♦ ❞❡ ♦✉tr❛✳

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●r❛❢♦s ❍♦♠❡♦♠♦r❢♦s

❉♦✐s ❣r❛❢♦s sã♦ ❤♦♠❡♦♠♦r❢♦s q✉❛♥❞♦ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ❞♦ ♠❡s♠♦ ❣r❛❢♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❞✐✈✐❞✐♥❞♦ ❛s ❛r❡st❛s ❞♦ ❣r❛❢♦ ♦r✐❣✐♥❛❧ ❛❞✐❝✐♦♥❛♥❞♦ ✈ért✐❝❡s✳

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❈❛♠✐♥❤♦s

• _{❯♠ ❝❛♠✐♥❤♦ ❡♠ ✉♠ ♠✉❧t✐❣r❛❢♦ é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✈ért✐❝❡s ❡} ❛r❡st❛s✱ ♥❛ q✉❛❧ ❝❛❞❛ ❛r❡st❛ei ❝♦♥té♠ ♦s ✈ért✐❝❡s vi−1 ❡ vi✿

v0, e1, v1, e2, v2, . . . , en−1, vn−1, en, vn.

• _{❯♠ ❝❛♠✐♥❤♦ s✐♠♣❧❡s é ✉♠ ❝❛♠✐♥❤♦ ♥♦ q✉❛❧ t♦❞♦s ♦s ✈ért✐❝❡s} sã♦ ❞✐st✐♥t♦s✳

• _{❯♠❛ tr✐❧❤❛ é ✉♠ ❝❛♠✐♥❤♦ ♥♦ q✉❛❧ t♦❞❛s ❛s ❛r❡st❛s sã♦ ❞✐st✐♥t❛s✳}

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❈❛♠✐♥❤♦s

• _{❚❡♦r❡♠❛ s♦❜r❡ ♣❛ss❡✐♦s ❡ ❝❛♠✐♥❤♦s s✐♠♣❧❡s✿ ❡①✐st❡ ✉♠} ❝❛♠✐♥❤♦ ❞❡ ✉♠ ✈ért✐❝❡ u ❛ ✉♠ ✈ért✐❝❡v s❡ ❡ s♦♠❡♥t❡ s❡ ❡①✐st❡

✉♠ ❝❛♠✐♥❤♦ s✐♠♣❧❡s ❞❡u ♣❛r❛ v✳

• _{❖✉tr❛ ♥♦♠❡♥❝❧❛t✉r❛ ✏♣❛ss❡✐♦✑ ❂ ❝❛♠✐♥❤♦✱ ✏❝❛♠✐♥❤♦✑ ❂ ❝❛♠✐♥❤♦} s✐♠♣❧❡s✳

• _{❖ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❝❛♠✐♥❤♦ é ♦ ♥ú♠❡r♦} n ❞❡ ❛r❡st❛s✳

• _{❙❡ ♥ã♦ ❤♦✉✈❡r ❛♠❜✐❣✉✐❞❛❞❡✱ é ♣♦ssí✈❡❧ ❞❡♥♦t❛r ♦ ❝❛♠✐♥❤♦ ♣♦r} ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✈ért✐❝❡s✳

• _{❖ ❝❛♠✐♥❤♦ é ❢❡❝❤❛❞♦ s❡} v0 =vn✳

• _{❯♠ ❝✐❝❧♦✱ ♦✉}k✲❝✐❝❧♦✱ é ✉♠ ❝❛♠✐♥❤♦ ❢❡❝❤❛❞♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ k≥3 ♥♦ q✉❛❧ t♦❞♦s ♦s ✈ért✐❝❡s sã♦ ❞✐st✐♥t♦s ❡①❝❡t♦ v0 =v_n✳

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❊①❡r❝í❝✐♦ ■

✶✳ ❚♦❞♦s ♦s ❝❛♠✐♥❤♦s✳ ❋❛ç❛ ✉♠❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s s✐♠♣❧❡s ❝♦♠ ❡①❛t❛♠❡♥t❡ ✹ ✈ért✐❝❡s ♥♦ ❣r❛❢♦ ❞❡✜♥✐❞♦ ♣❡❧♦s ❛r❝♦s ✼✲✸ ✶✲✹ ✼✲✽ ✵✲✺ ✺✲✷ ✸✲✽ ✷✲✾ ✵✲✻ ✹✲✾ ✷✲✻ ✻✲✹✳ ✭❢♦♥t❡✮

✷✳ ❚♦❞♦s ♦s ❝❛♠✐♥❤♦s✳ ❋❛ç❛ ✉♠❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s

s✐♠♣❧❡s ❝♦♠ ❡①❛t❛♠❡♥t❡ ✹ ✈ért✐❝❡s ♥♦ ❣r❛❢♦ ♥ã♦✲❞✐r✐❣✐❞♦ ❞❡✜♥✐❞♦ ♣❡❧❛s ❛r❡st❛s ✸✲✼ ✶✲✹ ✼✲✽ ✵✲✺ ✺✲✷ ✸✲✽ ✷✲✾ ✵✲✻ ✹✲✾ ✷✲✻ ✻✲✹✳ ✭❢♦♥t❡✮

✸✳ ❱❡r✐✜❝❛ ♣❛ss❡✐♦✳ ❊s❝r❡✈❛ ✉♠❛ ❢✉♥çã♦ ❜♦♦❧❡❛♥❛

●❘❆P❍❝❤❡❝❦P❛t❤ q✉❡ ✈❡r✐✜q✉❡ s❡ ✉♠❛ ❞❛❞❛ s❡q✉ê♥❝✐❛ s❡q❬✵✳✳❦❪ ❞❡ ✈ért✐❝❡s ❞❡ ✉♠ ❣r❛❢♦ é ✉♠ ♣❛ss❡✐♦✳ ❋❛ç❛ ❞✉❛s ✈❡rsõ❡s ❞❛ ❢✉♥çã♦✿ ✉♠❛ s✉♣õ❡ q✉❡ ♦ ❣r❛❢♦ é ❞❛❞♦ ♣♦r s✉❛ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛s ❡ ♦✉tr❛ s✉♣õ❡ q✉❡ ♦ ❣r❛❢♦ é ❞❛❞♦ ♣♦r s✉❛s ❧✐st❛s ❞❡ ❛❞❥❛❝ê♥❝✐❛✳ ✭❢♦♥t❡✮

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❊①❡r❝í❝✐♦ ■■

✹✳ ❱❡r✐✜❝❛ ❝❛♠✐♥❤♦ s✐♠♣❧❡s✳ ❊s❝r❡✈❛ ✉♠❛ ❢✉♥çã♦

●❘❆P❍❝❤❡❝❦❙✐♠♣❧❡P❛t❤ q✉❡ ✈❡r✐✜q✉❡ s❡ ✉♠❛ s❡q✉ê♥❝✐❛ s❡q❬✵✳✳❦❪ ❞❡ ✈ért✐❝❡s ❞❡ ✉♠ ❣r❛❢♦ é ✉♠ ❝❛♠✐♥❤♦ s✐♠♣❧❡s✳ ❆ ❢✉♥çã♦ ❞❡✈❡ ❞❡✈♦❧✈❡r ✲✶ s❡ ❛ s❡q✉ê♥❝✐❛ ♥ã♦ é ✉♠ ❝❛♠✐♥❤♦✱ ✶ s❡ ❛ s❡q✉ê♥❝✐❛ é ✉♠ ❝❛♠✐♥❤♦ s✐♠♣❧❡s✱ ❡ ✵ s❡ ❛ s❡q✉ê♥❝✐❛ é ✉♠ ❝❛♠✐♥❤♦ ♥ã♦ s✐♠♣❧❡s✳ ❋❛ç❛ ❞✉❛s ✈❡rsõ❡s ❞❛ ❢✉♥çã♦✿ ✉♠❛ s✉♣õ❡ q✉❡ ♦ ❣r❛❢♦ é ❞❛❞♦ ♣♦r s✉❛ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛s ❡ ♦✉tr❛ s✉♣õ❡ q✉❡ ♦ ❣r❛❢♦ é ❞❛❞♦ ♣♦r ❧✐st❛s ❞❡ ❛❞❥❛❝ê♥❝✐❛✳✭❢♦♥t❡✮

✺✳ ❈❛♠✐♥❤♦ ✈❡rs✉s ❝❛♠✐♥❤♦ s✐♠♣❧❡s✳ ▼♦str❡ q✉❡ ✉♠ ❣r❛❢♦ t❡♠ ✉♠ ❝❛♠✐♥❤♦ s✐♠♣❧❡s ❝♦♠ ♦r✐❣❡♠ s ❡ tér♠✐♥♦ t s❡ ❡ s♦♠❡♥t❡ s❡ t❡♠ ✉♠ ❝❛♠✐♥❤♦ ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ s✐♠♣❧❡s✮ ❝♦♠ ♦r✐❣❡♠ s ❡ tér♠✐♥♦ t✳

✻✳ ❈❛♠✐♥❤♦s ✐♥✈❡rs♦s✳ ▼♦str❡ q✉❡ ❡♠ q✉❛❧q✉❡r ❣r❛❢♦ ♥ã♦✲❞✐r✐❣✐❞♦ ♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ❝❛♠✐♥❤♦ t❛♠❜é♠ é ✉♠ ❝❛♠✐♥❤♦✳ ✭❢♦♥t❡✮

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❊①❡r❝í❝✐♦ ■■■

✼✳ ❱❡r✐✜❝❛ ❝✐❝❧♦✳ ❊s❝r❡✈❛ ✉♠❛ ❢✉♥çã♦ ❜♦♦❧❡❛♥❛ q✉❡ ✈❡r✐✜q✉❡ s❡ ✉♠❛ s❡q✉ê♥❝✐❛ s❡q❬✵✳✳❦❪ ❞❡ ✈ért✐❝❡s ❞❡ ✉♠ ❣r❛❢♦ é ✉♠ ❝✐❝❧♦✳ ❋❛ç❛ ❞✉❛s ✈❡rsõ❡s ❞❛ ❢✉♥çã♦✿ ✉♠❛ s✉♣õ❡ q✉❡ ♦ ❣r❛❢♦ é ❞❛❞♦ ♣♦r s✉❛ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛s ❡ ♦✉tr❛ s✉♣õ❡ q✉❡ ♦ ❣r❛❢♦ é ❞❛❞♦ ♣♦r ❧✐st❛s ❞❡ ❛❞❥❛❝ê♥❝✐❛✳ ✭❢♦♥t❡✮

✽✳ ❋❛ç❛ ✉♠❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s ❝✐❝❧♦s s✐♠♣❧❡s ✭❞✐❢❡r❡♥t❡s ❡♥tr❡ s✐✮ ♥♦ ❣r❛❢♦ ❞❡✜♥✐❞♦ ♣❡❧♦s ❛r❝♦s ✵✲✺ ✺✲✹ ✼✲✽ ✹✲✸ ✵✲✷ ✾✲✶✶ ✵✲✶ ✶✶✲✶✷ ✸✲✺ ✾✲✶✷ ✾✲✶✵ ✶✵✲✾ ✻✲✹ ✵✲✻✳✭❢♦♥t❡✮

✾✳ ❋❛ç❛ ✉♠❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s ❞✐❢❡r❡♥t❡s ❝✐❝❧♦s s✐♠♣❧❡s ♥♦ ❣r❛❢♦ ❞❡✜♥✐❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❝♦s ✵✲✹ ✶✲✵ ✷✲✶ ✷✲✸ ✸✲✵ ✸✲✹ ✹✲✶ ✹✲✷✳ ❋❛ç❛ ✉♠❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s ❝✐❝❧♦s s✐♠♣❧❡s ♥♦ ❣r❛❢♦ ❞❡✜♥✐❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❝♦s ✵✲✷ ✶✲✵ ✷✲✶ ✷✲✸ ✸✲✵ ✸✲✶✳ ✭❢♦♥t❡✮

✶✵✳ ❋❛ç❛ ✉♠❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s ❝✐❝❧♦s s✐♠♣❧❡s ♥♦ ❣r❛❢♦ ❞❡✜♥✐❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❝♦s ✵✲✷ ✶✲✵ ✷✲✶ ✷✲✸ ✸✲✵ ✸✲✶✳ ✭❢♦♥t❡✮

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❊①❡r❝í❝✐♦ ■❱

✶✶✳ ❋❛ç❛ ✉♠❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s ❝✐❝❧♦s s✐♠♣❧❡s ♥♦ ❣r❛❢♦ ♥ã♦✲❞✐r✐❣✐❞♦ ❞❡✜♥✐❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ✵✲✶ ✶✲✷ ✷✲✸✳ ✭❢♦♥t❡✮

✶✷✳ ❈✐❝❧♦s ✈❡rs✉s ❝✐❝❧♦s s✐♠♣❧❡s✳ ❙❡❥❛ ❛ ✉♠ ❛r❝♦ ❞❡ ✉♠ ❣r❛❢♦✳ ▼♦str❡ q✉❡ ❛ ♣❡rt❡♥❝❡ ❛ ✉♠ ❝✐❝❧♦ s❡ s♦♠❡♥t❡ s❡ ❛ ♣❡rt❡♥❝❡ ❛ ✉♠ ❝✐❝❧♦ s✐♠♣❧❡s✳ ✭❢♦♥t❡✮

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❈♦♥❡❝t✐✈✐❞❛❞❡

• _{❯♠ ❣r❛❢♦ é ❝♦♥❡①♦ s❡ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ❡♥tr❡ q✉❛✐sq✉❡r ❞♦✐s} ❞♦s s❡✉s ✈ért✐❝❡s✳

• H é ✉♠ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①♦ ❞❡ G s❡✿

• H é ✉♠ s✉❜❣r❛❢♦ ❝♦♥❡①♦ ❞❡G✱

• H ♥ã♦ é s✉❜❣r❛❢♦ ❞❡ ♥❡♥❤✉♠ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①♦ ❞❡ G✳ • _{❯♠ ❣r❛❢♦ ♣♦❞❡ s❡r ♣❛rt✐❝✐♦♥❛❞♦ ❡♠ s❡✉s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①♦s✳} • _{❚♦❞♦ ✈ért✐❝❡ ✐s♦❧❛❞♦ é ✉♠ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①♦✳ ❯♠ ✈ért✐❝❡} v é

✐s♦❧❛❞♦ s❡ deg(v) = 0✳

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❉✐stâ♥❝✐❛ ❡ ❉✐â♠❡tr♦

• _{❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ✈ért✐❝❡s}u ❡ v✱ ❞❡♥♦t❛❞❛ ♣♦r ✭✳u, v)✱ é ♦

❝♦♠♣r✐♠❡♥t♦ ❞♦ ♠❡♥♦r ❝❛♠✐♥❤♦ ❡♥tr❡ ❡st❡s ✈ért✐❝❡s✳

• _{❖ ❞✐â♠❡tr♦ ❞❡ ✉♠ ❣r❛❢♦}G✱ ❞❡♥♦t❛❞♦ ♣♦r ❞✐❛♠(G)✱ é ❛ ♠❛✐♦r

❞❛s ❞✐stâ♥❝✐❛s ❡♠ G✳

• _{❊①❡♠♣❧♦✿ ❞✐❛♠(}G) = 3,❞✐❛♠(H) = 4✳

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❈♦rt❡s ❡ ❈♦♥❡①õ❡s

• _❙❡❥❛G ✉♠ ❣r❛❢♦ ❝♦♥❡①♦✳

• _{❯♠ ✈ért✐❝❡}v é ✉♠ ❝♦rt❡ s❡ r❡♠♦✈❡♥❞♦ ❡st❡ ✈ért✐❝❡ ♦ ❣r❛❢♦

t♦r♥❛✲s❡ ❞❡s❝♦♥❡①♦✳

• _{❯♠❛ ❛r❡st❛} e é ✉♠❛ ❝♦♥❡①ã♦ s❡ r❡♠♦✈❡♥❞♦ ❡st❛ ❛r❡st❛ ♦ ❣r❛❢♦

t♦r♥❛✲s❡ ❞❡s❝♦♥❡①♦✳

• _{❊①❡♠♣❧♦✿} D é ✉♠ ❝♦rt❡ ❡♠ G❀ e={D, F} é ✉♠❛ ❝♦♥❡①ã♦ ❡♠ H✳

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❊①❡r❝í❝✐♦ ■

✶✳ ❋❛ç❛ ✉♠❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s s✉❜❣r❛❢♦s ❝♦♥❡①♦s ❞♦ ❣r❛❢♦ ❞❡✜♥✐❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ✵✲✶ ✵✲✷ ✵✲✸ ✶✲✸ ✷✲✸✳ ◗✉❛✐s ❞❡ss❡s s✉❜❣r❛❢♦s sã♦ ✐♥❞✉③✐❞♦s❄ ✭❢♦♥t❡✮

✷✳ ❙❡❥❛r ✉♠ ✈ért✐❝❡ ❞❡ ✉♠ ❣r❛❢♦✳ ❙✉♣♦♥❤❛ q✉❡ t♦❞♦ ✈ért✐❝❡ v ❞♦

❣r❛❢♦ é tér♠✐♥♦ ❞❡ ✉♠ ❝❛♠✐♥❤♦ ❝♦♠ ♦r✐❣❡♠r✳ ▼♦str❡ q✉❡ ♦

❣r❛❢♦ é ❝♦♥❡①♦✳ ✭❢♦♥t❡✮

✸✳ Pr♦♣r✐❡❞❛❞❡ ❞♦ ❝♦rt❡ ✈❛③✐♦✳ Pr♦✈❡ q✉❡ ✉♠ ❣r❛❢♦ é ❞❡s❝♦♥❡①♦ s❡ ❡ s♦♠❡♥t❡ s❡ ❛❧❣✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲tr✐✈✐❛❧ ❞❡ s❡✉s ✈ért✐❝❡s é ✐s♦❧❛❞♦✳ ✭❢♦♥t❡✮

✹✳ ◆ú♠❡r♦ ❞❡ ❛r❡st❛s✳ ◗✉❛♥t❛s ❛r❡st❛s t❡♠ ✉♠ ❣r❛❢♦ ❝♦♥❡①♦ ❝♦♠ ❱ ✈ért✐❝❡s❄ ✭❢♦♥t❡✮

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❊①❡r❝í❝✐♦ ■■

✺✳ ❊s❝r❡✈❛ ✉♠❛ ❢✉♥çã♦ ❯●❘❆P❍✐s❈♦♥♥❡❝t❡❞✭✮ q✉❡ ❞❡❝✐❞❛ s❡ ✉♠ ❣r❛❢♦ é ♦✉ ♥ã♦ é ❝♦♥❡①♦✳ ◆♦ ❝❛s♦ ❞❡ r❡s♣♦st❛ ♥❡❣❛t✐✈❛✱ q✉❡ ✐♥❢♦r♠❛çã♦ ❛❞✐❝✐♦♥❛❧ ❛ ❢✉♥çã♦ ♣♦❞❡r✐❛ ❞❡✈♦❧✈❡r ❝♦♠♦ ♣r♦✈❛ ❞❡ q✉❡ ❛ r❡s♣♦st❛ ❡stá ❝♦rr❡t❛❄ ✭❢♦♥t❡✮

✻✳ ❈❛♠✐♥❤♦s✳ ❊s❝r❡✈❛ ✉♠ ♣❡q✉❡♥♦ ♣r♦❣r❛♠❛ q✉❡ r❡❝❡❜❛ ✉♠ ❣r❛❢♦

G❡ ✉♠❛ ❧✐st❛ ❞❡ ♣❛r❡s ❞❡ ✈ért✐❝❡s ❡ ✐♠♣r✐♠❛✱ ♣❛r❛ ❝❛❞❛ ♣❛r

(s, t)✱ ✉♠ ❝❛♠✐♥❤♦ ❞❡ s ❛ t✳ ✭❢♦♥t❡✮

✼✳ ●r❛❢♦s ❝♦♥❡①♦s ❛❧❡❛tór✐♦s✳ ❊s❝r❡✈❛ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❝♦♥str✉✐r ❣r❛❢♦s ❛❧❡❛tór✐♦s ❝♦♥❡①♦s ❝♦♠V ✈ért✐❝❡s ❡ E ❛r❡st❛s✳ ❖ s❡✉

❛❧❣♦r✐t♠♦ ❞❡✈❡ ♣r♦❞✉③✐r ❝♦♠ ❛ ♠❡s♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ q✉❛❧q✉❡r ❞♦s ❣r❛❢♦s ❝♦♥❡①♦s ❝♦♠V ✈ért✐❝❡s ❡ E ❛r❡st❛s ✭❢♦♥t❡✮

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❊①❡r❝í❝✐♦ ■■■

✽✳ ▼♦str❡ q✉❡ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①❛ ❞❡ ✉♠ ❣r❛❢♦ G é ✉♠

s✉❜❣r❛❢♦ ❝♦♥❡①♦ ♠❛①✐♠❛❧ ❞❡ G✳ ❖ ❛❞❥❡t✐✈♦ ♠❛①✐♠❛❧ s✐❣♥✐✜❝❛ ♦

s❡❣✉✐♥t❡✿ s❡H é ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①❛ ❞❡ G❡♥tã♦ ♥ã♦ ❡①✐st❡ ✉♠

❣r❛❢♦ ❝♦♥❡①♦ H′ t❛❧ q✉❡_H _⊂_H′ _⊆_G✱ ♦✉ s❡❥❛✱ ♥ã♦ ❡①✐st❡ s✉❜❣r❛❢♦ ❝♦♥❡①♦ H′ ❞❡_G t❛❧ q✉❡_H é s✉❜❣r❛❢♦ ♣ró♣r✐♦ ❞❡ _H′✳ ✭❢♦♥t❡✮

✾✳ ❉❡t❡r♠✐♥❡ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞♦ ❣r❛❢♦ ♥❞ ❞❡✜♥✐❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ✸✲✼ ✶✲✹ ✼✲✽ ✵✲✺ ✺✲✷ ✸✲✽ ✷✲✾ ✵✲✻ ✹✲✾ ✷✲✻ ✻✲✹✳ ✭❢♦♥t❡✮

✶✵✳ ▼♦str❡ q✉❡ t♦❞❛ ❛r❡st❛ ❞❡ ✉♠ ❣r❛❢♦ t❡♠ ❛♠❜❛s ❛s ♣♦♥t❛s ♥❛ ♠❡s♠❛ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①❛✳ ✭❢♦♥t❡✮

✶✶✳ ▼♦str❡ q✉❡ t♦❞♦ ✈ért✐❝❡ ❞❡ ✉♠ ❣r❛❢♦ ♣❡rt❡♥❝❡ ❛ ✉♠❛ ❡ ✉♠❛ só ❞❡ s✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s✳ ✭❢♦♥t❡✮

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❆s ♣♦♥t❡s ❞❡ ❑ö♥✐s❜❡r❣

➱ ♣♦ssí✈❡❧ ❢❛③❡r ✉♠ ♣❛ss❡✐♦ ✈✐s✐t❛♥❞♦ ❝❛❞❛ ♣♦♥t❡ ❛♣❡♥❛s ✉♠❛ ✈❡③❄

(24)

●r❛❢♦ ❊✉❧❡r✐❛♥♦

• _{❯♠ ♠✉❧t✐❣r❛❢♦ é ❞✐t♦ ❛tr❛✈❡ssá✈❡❧ s❡}

• _{❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ q✉❡ ✈✐s✐t❛ t♦❞♦s ♦s ✈ért✐❝❡s ❡ ❝❛❞❛ ❛r❡st❛ é} ✈✐s✐t❛❞❛ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③✳

• _{❚❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ tr✐❧❤❛ ❛tr❛✈❡ssá✈❡❧✳}

• _{❯♠ ❣r❛❢♦ só ♣♦❞❡ s❡r ❛tr❛✈❡ssá✈❡❧ s❡ s❡✉s ✈ért✐❝❡s ✐♥t❡r♠❡❞✐ár✐♦s} t✐✈❡r❡♠ ❣r❛✉ ♣❛r ✭✉♠ ✈ért✐❝❡ é ✐♥t❡r♠❡❞✐ár✐♦ s❡ ♥ã♦ é ♥❡♠ ♦ ♣r✐♠❡✐r♦ ♥❡♠ ♦ ú❧t✐♠♦ ❞♦ ♣❛ss❡✐♦✮✳

• _{●r❛❢♦ ❡✉❧❡r✐❛♥♦✿ ✉♠ ❣r❛❢♦ ❝♦♥❡①♦ ✜♥✐t♦ é ❡✉❧❡r✐❛♥♦ s❡ ❡} s♦♠❡♥t❡ s❡ ❝❛❞❛ ✈ért✐❝❡ t❡♠ ❣r❛✉ ♣❛r✳

• _{◗✉❛❧q✉❡r ❣r❛❢♦ ❝♦♥❡①♦ ✜♥✐t♦ ❝♦♠ ❞♦✐s ✈ért✐❝❡s í♠♣❛r❡s é}

❛tr❛✈❡ssá✈❡❧✳ ❯♠❛ tr✐❧❤❛ ❛tr❛✈❡ssá✈❡❧ ♣♦❞❡ ❝♦♠❡ç❛r ❡♠ q✉❛❧q✉❡r ✈ért✐❝❡ í♠♣❛r ❡ t❡r♠✐♥❛r ♥♦ ♦✉tr♦ ✈ért✐❝❡ í♠♣❛r✳

(25)

❆s ♣♦♥t❡s ❞❡ ❑ö♥✐s❜❡r❣

➱ ♣♦ssí✈❡❧ ❢❛③❡r ✉♠ ♣❛ss❡✐♦ ✈✐s✐t❛♥❞♦ ❝❛❞❛ ♣♦♥t❡ ❛♣❡♥❛s ✉♠❛ ✈❡③❄

◆➹❖✦

(26)

●r❛❢♦ ❍❛♠✐❧t♦♥✐❛♥♦

• _{❈✐r❝✉✐t♦ ❤❛♠✐❧t♦♥✐❛♥♦✿ ❝❛♠✐♥❤♦ ❢❡❝❤❛❞♦ q✉❡ ✈✐s✐t❛ t♦❞♦s ♦s} ✈ért✐❝❡s ❞♦ ❣r❛❢♦ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③ ✭♣♦❞❡ r❡♣❡t✐r ❛r❡st❛s✮✳ ➱ ✉♠ ❝✐❝❧♦✳

• _{❈✐r❝✉✐t♦ ❡✉❧❡r✐❛♥♦✿ ♣❡r❝♦rr❡ ❝❛❞❛ ❛r❡st❛ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③✱} ♣♦❞❡ r❡♣❡t✐r ✈ért✐❝❡s✳

• _{❯♠ ❣r❛❢♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❛❞♠✐t❡ ✉♠ ❝✐❝❧♦ ❤❛♠✐❧t♦♥✐❛♥♦ ♣♦r} t♦❞♦s ♦s s❡✉s ✈ért✐❝❡s✳

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●r❛❢♦s r♦t✉❧❛❞♦s ❡ ♣♦♥❞❡r❛❞♦s

• _{❯♠ ❣r❛❢♦ r♦t✉❧❛❞♦ ❝♦♥té♠ ❞❛❞♦s ❛ss♦❝✐❛❞♦s ❛ s❡✉s ✈ért✐❝❡s} ❡✴♦✉ s✉❛s ❛r❡st❛s✳

• _{❯♠ ❣r❛❢♦ ♣♦♥❞❡r❛❞♦ ❝♦♥té♠ ✉♠ ♣❡s♦ ✭♦✉ ❝♦♠♣r✐♠❡♥t♦✮ ♥ã♦} ♥❡❣❛t✐✈♦ w(e)❛ss♦❝✐❛❞♦ ❛ ❝❛❞❛ ❛r❡st❛ e✳

• _{❖ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❝❛♠✐♥❤♦ ❡♠ ✉♠ ❣r❛❢♦ ♣♦♥❞❡r❛❞♦ é ❛} s♦♠❛ ❞♦s ♣❡s♦s ❞❛ ❛r❡st❛s ❞♦ ❝❛♠✐♥❤♦✳

• _{❖ ♠❡♥♦r ❝❛♠✐♥❤♦ ❡♥tr❡ ❞♦✐s ✈ért✐❝❡ ❡♠ ✉♠ ❣r❛❢♦ ♣♦♥❞❡r❛❞♦ é} ✉♠ ❝❛♠✐♥❤♦ ❝♦♠ ♦ ♠❡♥♦r ❝♦♠♣r✐♠❡♥t♦ ♣♦ssí✈❡❧✳

✭❊①✳✿P, A1, A2, A5, A3, A6, Q✮

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●r❛❢♦ ❝♦♠♣❧❡t♦

• _{❯♠ ❣r❛❢♦ é ❝♦♠♣❧❡t♦ s❡ ❝❛❞❛ ✈ért✐❝❡ ❡stá ❧✐❣❛❞♦ ❛ t♦❞♦s ♦s} ♦✉tr♦s✳

• _{❯♠ ❣r❛❢♦ ❝♦♠♣❧❡t♦ ❝♦♠}n ✈ért✐❝❡s é ❞❡♥♦t❛❞♦ ♣♦rKn✳

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●r❛❢♦ r❡❣✉❧❛r

• _❯♠k✲r❡❣✉❧❛r é ❛q✉❡❧❡ ♥♦ q✉❛❧ t♦❞❛s ♦s ✈ért✐❝❡s tê♠ ♦ ♠❡s♠♦

❣r❛✉ k✳

• _{❯♠ ❣r❛❢♦ ✷✲r❡❣✉❧❛r ❝♦♠} n ✈ért✐❝❡s é ❝❤❛♠❛❞♦ n✲❝✐❝❧♦✳

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●r❛❢♦ r❡❣✉❧❛r

• _{❖ ❣r❛❢♦ ✵✲r❡❣✉❧❛r é ♦ ❣r❛❢♦ tr✐✈✐❛❧✳} • _{❖ ❣r❛❢♦ ✶✲r❡❣✉❧❛r é}K2✳

• Kn é ✉♠ ❣r❛❢♦ ✭n−1✮✲r❡❣✉❧❛r✳

• _{❖s ❣r❛❢♦s ✸✲r❡❣✉❧❛r❡s tê♠ ✉♠ ♥ú♠❡r♦ ♣❛r ❞❡ ✈ért✐❝❡s✳} • _{❊①✐st❡♠ ❛♣❡♥❛s ✶✾ ❣r❛❢♦s ✸✲r❡❣✉❧❛r❡s ❝♦♠ ✶✵ ✈ért✐❝❡s✳}

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●r❛❢♦s ❜✐♣❛rt✐❞♦s

• _{❯♠ ❣r❛❢♦} G(V, E)é ❜✐♣❛rt✐❞♦ s❡ s❡✉ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s V

♣♦❞❡ s❡r ♣❛rt✐❝✐♦♥❛❞♦ ❡♠ ❞♦✐s ❝♦♥❥✉♥t♦sM ❡ N ❞❡ ♠♦❞♦ q✉❡

❝❛❞❛ ❛r❡st❛ ❞❡ G ❝♦♥❡❝t❛ ✉♠ ✈ért✐❝❡ ❞❡M ❛ ♦✉tr♦ ❞❡ N✳

• _{❊♠ ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ ❝♦♠♣❧❡t♦ ❝❛❞❛ ✈ért✐❝❡ ❞❡} M s❡ ❧✐❣❛ ❛

t♦❞♦s ♦s ✈ért✐❝❡s ❞❡N✳

• Km,n✱ é ♦ ❣r❛❢♦ ❜✐♣❛rt✐❞♦s ❝♦♠♣❧❡t♦ ❝♦♠ m ✈ért✐❝❡s ❡♠ M ❡ n

✈ért✐❝❡s ❡♠ N✳ P❛❞r♦♥✐③❛✲s❡m ≤n✳ Km,n t❡♠ mn ❛r❡st❛s✳

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➪r✈♦r❡s

• _{❯♠ ár✈♦r❡s é ✉♠ ❣r❛❢♦ ❝♦♥❡①♦ s❡♠ ❝✐❝❧♦s✳} • _{❯♠❛ ✢♦r❡st❛ é ✉♠ ❣r❛❢♦ s❡♠ ❝✐❝❧♦s✳}

• _{❖s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①♦s ❞❛ ✢♦r❡st❛ sã♦ ár✈♦r❡s✳}

• _{❯♠❛ ár✈♦r❡s ❞❡❣❡♥❡r❛❞❛ t❡♠ ✉♠ ú♥✐❝♦ ✈ért✐❝❡ ❡ ♥❡♥❤✉♠❛ ❛r❡st❛✳}

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❖s ❝❛♠✐♥❤♦s ♥❛s ár✈♦r❡s

❙❡❥❛T ✉♠❛ ár✈♦r❡✳

• _{❊①✐st❡ ✉♠ ú♥✐❝♦ ❝❛♠✐♥❤♦ s✐♠♣❧❡s ❡♥tr❡ ❞♦✐s ✈ért✐❝❡s ❞❡} T✳

• _{❙❡ ❡①✐st✐ss❡ ♠❛✐s ❞❡ ✉♠ ❝❛♠✐♥❤♦ s✐♠♣❧❡s s❡r✐❛ ♣♦ssí✈❡❧ ❢♦r♠❛r} ✉♠ ❝✐❝❧♦✳

• _{❙❡ ❝♦♥str✉✐r♠♦s ✉♠ ❣r❛❢♦} T′ ❛❞✐❝✐♦♥❛♥❞♦ ❛_T ✉♠❛ ♥♦✈❛ ❛r❡st❛

e={u, v}✱ ✈❛✐ ❡①✐st✐r ♠❛✐s ❞❡ ✉♠ ❝❛♠✐♥❤♦ s✐♠♣❧❡s ❡♥tr❡ u ❡v✱

❧♦❣♦ T′ ♥ã♦ é ✉♠❛ ár✈♦r❡✳

• _{❙❡ ❝♦♥str✉✐r♠♦s ✉♠ ❣r❛❢♦} T′′ r❡♠♦✈❡♥❞♦ ❞❡ _T ❛ ❛r❡st❛

e={u, v}✱ ♥ã♦ ✈❛✐ ♠❛✐s ❡①✐st✐r ✉♠ ❝❛♠✐♥❤♦ s✐♠♣❧❡s ❡♥tr❡ u ❡ v✱

T′′ é ❞❡s❝♦♥❡①♦✱ ❧♦❣♦ ♥ã♦ é ✉♠❛ ár✈♦r❡✳ • T′′ é ✉♠❛ ✢♦r❡st❛❄

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n

✈ért✐❝❡s ❡

n

−

1 ❛r❡st❛s

❚♦❞❛ ár✈♦r❡ ❝♦♠ n ✈ért✐❝❡s t❡♠ ❡①❛t❛♠❡♥t❡ n−1 ❛r❡st❛s✳ Pr♦✈❛ ♣♦r ✐♥❞✉çã♦✿

• _{❱❡r❞❛❞❡ ♣❛r❛} n= 1,2❡ ✸✳

• _{❙✉♣♦♥❤❛ q✉❡ é ✈❡r❞❛❞❡ ♣❛r❛}n < n0✳ • _❙❡❥❛T0 ✉♠❛ ár✈♦r❡ ❝♦♠ n0 ✈ért✐❝❡s✳

• _{❘❡♠♦✈❛ ✉♠❛ ❛r❡st❛ ❞❡} T ❣❡r❛♥❞♦ ❞✉❛s ár✈♦r❡s T1 ❡ T2✱ ❝♦♠ n1 ❡n2 ♥ós r❡s♣❡❝t✐✈❛♠❡♥t❡✳

• T1 t❡♠ n1−1 ❛r❡st❛s ❡T2 t❡♠n2−1 ❛r❡st❛s✳

• _▲♦❣♦T0 t❡♠1 + (n1−1) + (n2−1) =n0 −1 ❛r❡st❛s✱ ❧❡♠❜r❡ q✉❡n0 =n1+n2✳

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➪r✈♦r❡s ●❡r❛❞♦r❛s

• _❙❡❥❛G ✉♠ ❣r❛❢♦ ❝♦♥❡①♦✳

• _❙❡❥❛T ✉♠ s✉❜❣r❛❢♦ ❞❡ G✳

• _❙❡T é ✉♠❛ ár✈♦r❡ ❡ ✐♥❝❧✉✐ t♦❞♦s ♦s ✈ért✐❝❡s ❞❡G✱ ❡♥tã♦ T é

✉♠❛ ár✈♦r❡ ❣❡r❛❞♦r❛✳

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➪r✈♦r❡ ❣❡r❛❞♦r❛ ♠í♥✐♠❛

• _{❯♠❛ ár✈♦r❡ ❣❡r❛❞♦r❛ ♠í♥✐♠❛ ❡♠ ✉♠ ❣r❛❢♦ ❝♦♥❡①♦ ❝♦♠ ♣❡s♦} é ❛ ár✈♦r❡ ❣❡r❛❞♦r❛ ❝✉❥❛ s♦♠❛ ❞♦s ♣❡s♦s ❞❛s ❛r❡st❛s é ❛ ♠❡♥♦r ♣♦ssí✈❡❧✳

• _{❆❧❣♦r✐t♠♦ ❞❡❧❡çã♦ ❞❡ ❛r❡st❛s ♣❛r❛ ❡♥❝♦♥tr❛r ár✈♦r❡ ❣❡r❛❞♦r❛} ♠í♥✐♠❛✿

✶✳ ❖r❞❡♥❡ ❛s ❛r❡st❛s ❞♦ ❣r❛❢♦ ❞♦ ♠❛✐♦r ♣❛r❛ ♦ ♠❡♥♦r ♣❡s♦✳

✷✳ ❙❡❣✉✐♥❞♦ ❛ ♦r❞❡♠ ❞❛s ❛r❡st❛s ❞❡❧❡t❡ ✈á ❞❡❧❡t❛♥❞♦ ❛s ❛r❡st❛s q✉❡ ♥ã♦ ❞❡s❝♦♥❡❝t❛♠ ♦ ❣r❛❢♦ ❛té q✉❡ r❡st❡♠ ❛♣❡♥❛s n₋₁✳

• _{❆❧❣♦r✐t♠♦ ❞❡ ❑r✉s❦❛❧✿}

✶✳ ❖r❞❡♥❡ ❛s ❛r❡st❛s ❞♦ ❣r❛❢♦ ❞♦ ♠❡♥♦r ♣❛r❛ ♦ ♠❛✐♦r ♣❡s♦✳

✷✳ ❆❞✐❝✐♦♥❡ ❛s ❛r❡st❛s ♥❛ ♦r❞❡♠ ❛té q✉❡ ❝♦♠♣❧❡t❡n₋₁ ❛r❡st❛s✱ só ❛❞✐❝✐♦♥❡ ❛q✉❡❧❛s ❛r❡st❛s q✉❡ ♥ã♦ ❣❡r❛♠ ❝✐❝❧♦s✳

• _{❯♠ ú♥✐❝♦ ❣r❛❢♦ ♣♦❞❡ t❡r ✈ár✐❛s ár✈♦r❡s ❣❡r❛❞♦r❛ ♠í♥✐♠❛s} ❞✐st✐♥t❛s✳

(37)

❆❧❣♦r✐t♠♦ ❞❡ ❞❡❧❡çã♦ ❞❡ ❛r❡st❛s

(38)

❆❧❣♦r✐t♠♦ ❞❡ ❑✉s❦❛❧

(39)

●r❛❢♦s ♣❧❛♥❛r❡s

• _{❯♠ ❣r❛❢♦ é ♣❧❛♥❛r q✉❛♥❞♦ é ♣♦ssí✈❡❧ ❞❡s❡♥❤á✲❧♦ s❡♠ q✉❡ s✉❛s} ❛r❡st❛s s❡ ❝r✉③❡♠✳

• _{❚♦❞❛ ár✈♦r❡ é ♣❧❛♥❛r✳} • _{❊①❡♠♣❧♦✿ ❣r❛❢♦} K4✳

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▼❛♣❛s ❡ ❘❡❣✐õ❡s

• _{❯♠ ♠✉❧t✐❣r❛❢♦ ♣❧❛♥❛r ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ✉♠ ♠❛♣❛} q✉❡ ❞✐✈✐❞❡ ♦ ♣❧❛♥♦ ❡♠ r❡❣✐õ❡s✳

• _{❆s ❛r❡st❛s ❞❡✜♥❡♠ ❞❛s ❜♦r❞❛s ❞❡ ✉♠❛ r❡❣✐ã♦✳} • _{❊st❛s ❜♦r❞❛s ♣♦❞❡♠ ♦✉ ♥ã♦ ❢♦r♠❛r ✉♠ ❝✐❝❧♦✳}

• _{❊①❡♠♣❧♦✿ ❛s ❜♦r❞❛s ❞❡} r3 ♥ã♦ ❢♦r❛♠ ✉♠ ❝✐❝❧♦✱ ♣♦✐s ❛ ❛r❡st❛

{E, F} s❡ r❡♣❡t❡ ♥❛ ❜♦r❞❛ (C, D, E, F, E, C).

(41)

●r❛✉ ❞❛ r❡❣✐ã♦

• _{❖ ❣r❛✉ ❞❡ ✉♠❛ r❡❣✐ã♦} r✱ ❞❡♥♦t❛❞♦ ♣♦r deg(r)✱ é ♦

❝♦♠♣r✐♠❡♥t♦ ❞♦ ❝❛♠✐♥❤♦ ❞❛ ❜♦r❞❛ ❞❡ r✳

• _{❚❡♦r❡♠❛ s♦❜r❡ ♦s ❣r❛✉s ❞❛s r❡❣✐õ❡s ❡ ♦ ♥ú♠❡r♦ ❞❡} ❛r❡st❛s ❞♦ ❣r❛❢♦✿ ❛ s♦♠❛ ❞♦s ❣r❛✉s ❞❛s r❡❣✐õ❡s ❞❡ ✉♠ ♠❛♣❛ é ✐❣✉❛❧ ❛ ❞✉❛s ✈❡③❡s ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✳

(42)

❋ór♠✉❧❛ ❞❡ ❊✉❧❡r

• _{❆ss♦❝✐❛ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s} p✱ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q ❡ ♦ ♥ú♠❡r♦

❞❡ r❡❣✐õ❡s r ❡♠ ✉♠ ❣r❛❢♦ ♣❧❛♥❛r ❝♦♥❡①♦ ✭♠❛♣❛✮✿

p−q−r = 2.

• _{❊st❡ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦ ♣❛r❛ ár✈♦r❡s❄} • _{❊①❡♠♣❧♦ ❛❜❛✐①♦✿} p−q+r= 6−9 + 5.

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●r❛✉ ♠í♥✐♠♦ ❞❡ ✉♠❛ r❡❣✐ã♦

• _❙❡❥❛G ✉♠ ♠✉❧t✐❣r❛❢♦ ❝♦♥❡①♦ ♣❧❛♥❛r ❝♦♠ ✸ ♦✉ ♠❛✐s ✈ért✐❝❡s

✭♥ã♦ éK1 ♥❡♠ K2✮✳

• _❙❡❥❛M ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥❛r ❞❡ G✿ ✶✳ ❯♠❛ r❡❣✐ã♦ ❞❡ ❣r❛✉ ✶ é ✉♠ ❧❛ç♦✳

✷✳ ❯♠❛ r❡❣✐ã♦ ❞❡ ❣r❛✉ ✷ é ❝♦♠♣♦rt❛ ♣♦r ❛r❡st❛s ♠ú❧t✐♣❧❛s✳

✸✳ ❆s ❞❡♠❛✐s r❡❣✐õ❡s t❡♠ ❣r❛✉ ♠❛✐♦r ✐❣✉❛❧ ❛ ✸✳

• _{❊♠ ✉♠ ❣r❛❢♦ s✐♠♣❧❡s ✭♥ã♦ é ♠✉❧t✐❣r❛❢♦✮ ❝♦♥❡①♦ ♣❧❛♥❛r✱ t♦❞❛s ❛s} ❛r❡st❛s t❡♠ ❣r❛✉ ♠❛✐♦r ✐❣✉❛❧ ❛ ✸✳

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◆ú♠❡r♦ ♠á①✐♠♦ ❞❡ ❛r❡st❛s ❡♠ ✉♠ ❣r❛❢♦ ♣❧❛♥❛r

❝♦♥❡①♦

❙❡❥❛G ✉♠ ❣r❛❢♦ ❝♦♥❡①♦ ♣❧❛♥❛r ❝♦♠p ✈ért✐❝❡s ❡ q ❛r❡st❛s✱ ❝♦♠ p≥3✱ t❡♠♦s✿

q ≤3p−6.

• _{❆ s♦♠❛ ❞♦s ❣r❛✉s ❞❛s r❡❣✐õ❡s é ❞✉❛s ✈❡③❡s ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✱} 2q✳

• _{❖ ❣r❛✉ ❞❡ ❝❛❞❛ r❡❣✐ã♦ é ♥♦ ♠í♥✐♠♦ ✸✳}

2q ≥3r ❧♦❣♦ r≥2q/3.

• _{❙✉❜st✐t✉✐♥❞♦ ♥❛ ❢ór♠✉❧❛ ❞❡ ❊✉❧❡r✿}

2 =p−q+r ≤p−q+2q

3 ♦✉ 2≤p−

q

3.

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K

3,3

é ♥ã♦ ♣❧❛♥❛r

• _{●r❛❢♦ ❜✐♣❛rt✐❞♦ ❝♦♠♣❧❡t♦ ❝♦♠ ✸ ✈ért✐❝❡s ❡♠ ❝❛❞❛ ❧❛❞♦✳} • p= 6 ✈ért✐❝❡s ❡ q= 9 ❛r❡st❛s✳

❙❡ ❢♦ss❡ ♣❧❛♥❛r✿

• _{P❡❧❛ ❢ór♠✉❧❛ ❞❡ ❊✉❧❡r t❡r✐❛ ✺ r❡❣✐õ❡s✳}

• _{■♠♣♦ssí✈❡❧ t❡r r❡❣✐õ❡s ❞❡ ❣r❛✉ ✸ ♣♦r s❡r ❜✐♣❛rt✐❞♦✳} • _{❚♦❞❛ r❡❣✐ã♦ t❡♠ ♥♦ ♠í♥✐♠♦ ❣r❛✉ ✹✳}

• _{❆ s♦♠❛ ❞♦ ❣r❛✉s ❞❛ r❡❣✐õ❡s é ♠❛✐♦r ✐❣✉❛❧ ❛ ✷✵✳}

• _{❖ ♥ú♠❡r♦ ❞❡ ❛r❡st❛ ❞❡✈❡ s❡r ♣❡❧♦ ♠❡♥♦r ✶✵✳ ✭❈♦♥tr❛❞✐çã♦✦✮}

(46)

❚❡♦r❡♠❛ ❞❡ ❑✉r❛t♦✇s❦✐

K5 ♥ã♦ é ♣❧❛♥❛r✳

• K5 é ♦ ❣r❛❢♦ ❝♦♠♣❧❡t♦ ❝♦♠ p= 5 ✈ért✐❝❡s ❡ q= 10 ❛r❡st❛s✳ • _{❙❡ ❢♦ss❡ ✉♠ ❣r❛❢♦ ♣❧❛♥❛r ♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ ❛r❡st❛s s❡r✐❛}

q ≤3p−6 = 15−6 = 9 ✭❈♦♥tr❛❞✐çã♦✦✮.

❚❡♦r❡♠❛ ❞❡ ❑✉r❛t♦✇s❦✐

❯♠ ❣r❛❢♦ é ♥ã♦ ♣❧❛♥❛r s❡ ❡ s♦♠❡♥t❡ s❡ ❝♦♥té♠ ✉♠ s✉❜❣r❛❢♦ ❤♦♠❡♦♠♦r❢♦ ❛K3,3 ♦✉ K5✳

(47)

❈♦❧♦r❛çã♦ ❞❡ ●r❛❢♦s

• _{❯♠❛ ❝♦❧♦r❛çã♦ ❡♠ ✉♠ ❣r❛❢♦ é ✉♠❛ ❛tr✐❜✉✐çã♦ ❞❡ ❝♦r❡s ❛♦ s❡✉s} ✈ért✐❝❡s ❞❡ ♠♦❞♦ q✉❡ ♦s ✈ért✐❝❡s ❛❞❥❛❝❡♥t❡s tê♠ ❝♦r❡s ❞✐st✐♥t❛s✳ • _{❯♠ ❣r❛❢♦ é ❞✐t♦}k✲❝♦❧♦rá✈❡❧ s❡ ❡①✐st❡ ✉♠❛ ❝♦❧♦r❛çã♦ q✉❡ ✉t✐❧✐③❛ k

❝♦r❡s✳

• _{❖ ♥ú♠❡r♦ ❝r♦♠át✐❝♦ ❞❡ ✉♠ ❣r❛❢♦}G é ♦ ♠❡♥♦r ♥ú♠❡r♦ ❝♦r❡s

♥❡❝❡ssár✐♦ ♣❛r❛ ❢❛③❡r s✉❛ ❝♦❧♦r❛çã♦✱ ❞❡♥♦t❛❞♦ ♣♦r χ(G)✳

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❆❧❣♦r✐t♠♦ ❞❡ ❲❡❧❝❤✲P♦✇❡❧❧

• _{❆❧❣♦r✐t♠♦ ❞❡ ❲❡❧s❤✲P♦✇❡❧❧ ♣❛r❛ ❝♦❧♦r❛çã♦ ❞❡ ❣r❛❢♦s✳} • _{❙✐♠♣❧❡s ♠❛s ♥❡♠ s❡♠♣r❡ ❢♦r♥❡❝❡ ❛ ❝♦❧♦r❛çã♦ ♠í♥✐♠❛✳} ❊♥tr❛❞❛✿ ✉♠ ❣r❛❢♦ G✳

✶✳ ❖r❞❡♥❡ ♦s ✈ért✐❝❡s ❡♠ ♦r❞❡♠ ❞❡❝r❡s❝❡♥t❡ ❞♦s ❣r❛✉s✳

✷✳ ❋❛ç❛ i= 1.

✸✳ ❊♥q✉❛♥t♦ ❡①✐st✐r ✈ért✐❝❡s ♥ã♦ ❝♦❧♦r✐❞♦

✸✳✶ ❆ss♦❝✐❡ ❛ ❝♦ri❛♦ ♣ró①✐♠♦ ✈ért✐❝❡ ❞❛ ❧✐st❛ ❛✐♥❞❛ s❡♠ ❝♦r✳

✸✳✷ ❆ss♦❝✐❡ ❛ ❝♦ri❛♦ ♣ró①✐♠♦ ✈ért✐❝❡ ❞❛ ❧✐st❛ ❛✐♥❞❛ s❡♠ ❝♦r ♥ã♦ ❛❞❥❛❝❡♥t❡ ❛♦s ✈ért✐❝❡s ❝♦♠ ❝♦ri✳

✸✳✸ i_{+ +}.

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❊①❡♠♣❧♦

(50)

❊①❡♠♣❧♦

◆ã♦ é ✷✲❝♦❧♦rá✈❡❧ ♣♦rq✉❡ A1, A2 ❡ A3 ❡stã♦ ❝♦♥❡❝t❛❞♦s ❡♥tr❡ s✐✳

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❈♦❧♦r❛çõ❡s ♠á①✐♠❛✳

• χ(Kn) =n✳

• _{➱ ❡q✉✐✈❛❧❡♥t❡ ❞✐③❡r s♦❜r❡ ✉♠ ❣r❛❢♦}G✿

• Gé ✷✲❝♦❧♦rá✈❡❧✳ • Gé ❜✐♣❛rt✐❞♦✳

• _{❚♦❞♦ ❝✐❝❧♦ ❞❡}Gt❡♠ ❣r❛✉ í♠♣❛r✳ • _{◗✉❛❧q✉❡r ❣r❛❢♦ ♣❧❛♥❛r é ✺✲❝♦❧♦rá✈❡❧✳} • _{◗✉❛❧q✉❡r ❣r❛❢♦ ♣❧❛♥❛r é ✹✲❝♦❧♦rá✈❡❧ ✭❄✮✳}

(52)

▼❛♣❛ ❞✉❛❧

• _{❯t✐❧✐③❛❞♦ ♣❛r❛ ❞❡✜♥✐r ❛ ❝♦❧♦r❛çã♦ ❞❛s r❡❣✐õ❡s ❞❡ ✉♠ ♠❛♣❛✳} • _{❈❛❞❛ r❡❣✐ã♦ é ✉♠ ♠❛♣❛}M é r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠ ✈ért✐❝❡ ♥♦

♠❛♣❛ ❞✉❛❧ M∗✳

• _{❈❛❞❛ ❛r❡st❛ ❡♠}M∗ ❝r✉③❛♠ ❛s ❛r❡st❛s q✉❡ ❢❛③❡♠ ❢r♦♥t❡✐r❛s ❡♠ M ❝♦♥❡❝t❛♥❞♦ ❞✉❛s r❡❣✐õ❡s✳

(53)

❚❡♦r❡♠❛ ❞❛s q✉❛tr♦ ❝♦r❡s

• _{❆♣♣❡❧ ✫ ❑❛❦❡♥♥ ✭✶✾✼✻✮✳}

• _{❚♦❞♦ ❣r❛❢♦ ♣❧❛♥❛r é ✹✲❝♦❧♦rá✈❡❧✳}

• _{❙❡ ❛s r❡❣✐õ❡s ❞❡ q✉❛❧q✉❡r ♠❛♣❛} M sã♦ ❝♦❧♦r✐❞❛s ❞❡ ❢♦r♠❛ q✉❡

r❡❣✐õ❡s ✈✐③✐♥❤❛s tê♠ ❝♦r❡s ❞✐st✐♥t❛s✱ ❡♥tã♦ ♥ã♦ ♠❛✐s q✉❡ q✉❛tr♦ ❝♦r❡s sã♦ ♥❡❝❡ssár✐❛s✳

• _{❆ ♣r♦✈❛ ❞❡st❡ t❡♦r❡♠❛ ❞❡♣❡♥❞❡✉ ❞♦ ❛✉①í❧✐♦ ❞❡ ❝♦♠♣✉t❛❞♦r❡s} ♣❛r❛ ♠♦str❛r q✉❡ ♥ã♦ ❡①✐st✐❛ ✉♠ ❝♦♥tr❛✲❡①❡♠♣❧♦ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❝❛❧❝✉❧❛❞♦ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✷✵✵✵ ❣r❛❢♦s ♣❧❛♥❛r❡s✳

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❘❡♣r❡s❡♥t❛çã♦ ❞❡ ●r❛❢♦s

• p✈ért✐❝❡s ❡ q ❛r❡st❛s✳

• _{▼❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛✳}

• _{❯t✐❧✐③❛❞❛ ♣❛r❛ ❣r❛❢♦s ❞❡♥s♦s✳} • _{▲✐st❛ ❞❡ ❛❞❥❛❝ê♥❝✐❛✳}

• _{❯t✐❧✐③❛❞❛ ♣❛r❛ ❣r❛❢♦s ❡s♣❛rs♦s✳} • _{▼❛tr✐③ ❞❡ ✐♥❝✐❞ê♥❝✐❛✳}

• _{❈❛❞❛ r❡♣r❡s❡♥t❛çã♦ é ♠❛✐s ❛❞❡q✉❛❞❛ ❛ ✉♠ ❛❧❣♦r✐t♠♦✳}

• _{➱ ❝♦♠✉♠ ❝♦♠♣❛rt✐❧❤❛r ❣r❛❢♦s ❝♦♠♦ ❛rq✉✐✈♦s ❞❡ ❧✐st❛s ❞❡ ❛r❡st❛s✱} ♥♦ q✉❛❧ ❝❛❞❛ ❧✐♥❤❛ t❡♠ ✷ ♦✉ ✸ ❡❧❡♠❡♥t♦s✿ ❝❛❞❛ ✈ért✐❝❡ ✭❡ ♦ ♣❡s♦✮✳

(55)

▼❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛

• _{❘❡❧❛❝✐♦♥❛ ❝❛❞❛ ✈ért✐❝❡ ❝♦♠ t♦❞♦s ♦s ✈ért✐❝❡s ❞♦ ❣r❛❢♦✳} • _{❖ ✈❛❧♦r❡s ❞❛ ♠❛tr✐③ ♣♦❞❡♠ r❡♣r❡s❡♥t❛r✿}

• _{❆ ❡①✐stê♥❝✐❛ ❞❡ ❛r❡st❛s ❡♠ ✉♠ ❣r❛❢♦ s✐♠♣❧❡s✳} • _{❖ ♣❡s♦ ❡♠ ✉♠ ❣r❛❢♦ ♣♦♥❞❡r❛❞♦✳}

• _{❖ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❡♠ ✉♠ ♠✉❧t✐❣r❛❢♦✳}

(56)

▲✐st❛ ❞❡ ❛❞❥❛❝ê♥❝✐❛

• _{❚❡♠ ✉♠❛ ❧✐st❛ ♣❛r❛ ❝❛❞❛ ✈ért✐❝❡✳}

• _{❆ ❧✐st❛ ✐♥❞✐❝❛ ❛ q✉❛✐s ✈ért✐❝❡s ❡stá ❧✐❣❛❞♦ ♦ ✈ért✐❝❡ ✏❞♦♥♦✑ ❞❡st❛} ❧✐st❛✳

(57)

▼❛tr✐③ ❞❡ ✐♥❝✐❞ê♥❝✐❛

• _{❘❡❧❛❝✐♦♥❛ ❛r❡st❛s ❡ ✈ért✐❝❡s✳}

• _{▼❛r❝❛ ✶ q✉❛♥❞♦ ❛ ❛r❡st❛ ❡ ♦ ✈ért✐❝❡ ✐♥❝✐❞❡♠✳}

(58)

❘❡❢❡rê♥❝✐❛

✶✳ ❙✳ ▲✐♣s❝❤✉t③ ✫ ▼✳ ▲✳ ▲✐♣s♦♥✳ ❚❡♦r✐❛ ❡ Pr♦❜❧❡♠❛s ❞❡ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ✷❛ _{❡❞✐çã♦✳ ❊❞✐t♦r❛ ❇♦♦❦♠❛♥ ✭❈♦❧❡çã♦ ❙❝❤❛✉♠✮✱ ✷✵✵✹✳}

✷✳ ❁❤tt♣✿✴✴❝✐♦✳❝♦♠✳❜r✴ ❢❛❝❛✲❛✲t❡♦r✐❛✲❞♦s✲❣r❛❢♦s✲tr❛❜❛❧❤❛r✲♣❛r❛✲s✉❛✲❡♠♣r❡s❛✴❃ ✸✳ ❁❤tt♣s✿✴✴✇✇✇✳✐♠❡✳✉s♣✳❜r✴⑦♣❢✴❛❧❣♦r✐t♠♦s❴♣❛r❛❴❣r❛❢♦s✴❃ ✹✳ ❁❤tt♣s✿✴✴✇✇✇✳❤❡✐❞❡❧❜❡r❣✲❧❛✉r❡❛t❡✲❢♦r✉♠✳♦r❣✴❜❧♦❣✴ t❤❡✲❜r✐❞❣❡s✲♦❢✲❦♦♥✐❣s❜❡r❣✴❃ ✺✳ ❁❤tt♣✿✴✴❝♦♠♣❛❧❣✳✐♥❢✳❡❧t❡✳❤✉✴⑦t♦♥②✴❖❦t❛t❛s✴❚❉❑✴❋■◆❆▲✴ ❈❤❛♣✪✷✵✹✳P❉❋❃ ✻✳ ❁❤tt♣s✿✴✴✇✇✇✳✐❝✳✉♥✐❝❛♠♣✳❜r✴⑦❛t✐❧✐♦✴s❧✐❞❡s❲t✐s❝✳♣❞❢❃ ✼✳ ❁❤tt♣✿✴✴✇✇✇✳✐❜✐❧❝❡✳✉♥❡s♣✳❜r✴❍♦♠❡✴❉❡♣❛rt❛♠❡♥t♦s✴ ▼❛t❡♠❛t✐❝❛❆♣❧✐❝❛❞❛✴s♦❝♦rr♦✹✵✷✾✴r❡♣r❡s❡♥t❛❴❣r❛❢♦s✷✳♣❞❢❃