Rozklady propojovacích sítí na dlouhé cesty

Abstract

◆á③❡✈ ♣rá❝❡✿ ❘♦③❦❧❛❞② ♣r♦♣♦❥♦✈❛❝í❝❤ sítí ♥❛ ❞❧♦✉❤é ❝❡st② ❆✉t♦r✿ ❑r②➨t♦❢ ▼➙❦✉t❛ ❑❛t❡❞r❛✿ ❑❛t❡❞r❛ t❡♦r❡t✐❝❦é ✐♥❢♦r♠❛t✐❦② ❛ ♠❛t❡♠❛t✐❝❦é ❧♦❣✐❦② ❱❡❞♦✉❝í ❜❛❦❛❧á➦s❦é ♣rá❝❡✿ ▼❣r✳ P❡tr ●r❡❣♦r✱ P❤✳❉✳✱ ❑❚■▼▲ ❆❜str❛❦t✿ ❱ tét♦ ♣rá❝✐ ❥s♦✉ s❤r♥✉t② ✈❧❛st♥♦st✐ ✈②❜r❛♥ý❝❤ ♣r♦♣♦❥♦✈❛❝í❝❤ sítí✱ ❥✐♠✐➸ ❥s♦✉ ♠♦❞✐✜❦❛❝❡ ❤②♣❡r❦r②❝❤❧❡✳ ❑♦♥❦rét♥➙ st✉❞✉❥❡♠❡ t♦❧❡r❛♥❝✐ ❦ ❝❤②❜á♠ ✈ t③✈✳ ❛✉❣♠❡♥t♦✈❛♥ý❝❤ ❦r②❝❤❧í❝❤ s ♦❤❧❡❞❡♠ ♥❛ ❡①✐st❡♥❝✐ ❤❛♠✐❧t♦♥♦✈s❦ý❝❤ ❦r✉➸♥✐❝✳ ❚❛❦é ♣➦❡❞✈❡❞❡♠❡ ♠❡t♦❞✉ ♣♦✉➸í✈❛❥í❝í ❥❡❞♥♦❞✉❝❤é ♣r♦st➦❡❞❦② ❧✐♥❡ár♥í ❛❧❣❡❜r②✱ ♣♦♠♦❝í ❦t❡ré ❧③❡ ③❦♦♥str✉♦✈❛t ✈❡❧❦é ♠♥♦➸st✈í ♣♦❞❣r❛❢➲ n✲❞✐♠❡♥③✐♦♥á❧♥í ❛✉❣♠❡♥t♦✈❛♥é ❦r②❝❤❧❡ AQn ✐③♦♠♦r❢♥í❝❤ n✲❞✐♠❡♥③✐♦♥á❧♥í ❤②♣❡r❦r②❝❤❧✐ Qn✳ ❉♦❦á➸❡♠❡✱ ➸❡ AQn s f ✈❛❞♥ý♠✐ ❤r❛♥❛♠✐ ♦❜s❛❤✉❥❡ ❦♦♣✐✐ Qn s ♥❡❥✈ý➨❡ n 2n−1 f ✈❛❞♥ý♠✐ ❤r❛♥❛♠✐✳ P♦♠♦❝í t♦❤♦t♦ ✈ýs❧❡❞❦✉ ❥s♠❡ s❝❤♦♣♥✐ ♥➙❦t❡ré ✈❧❛st♥♦st✐ Qn s ✈❛❞♥ý♠✐ ❤r❛♥❛♠✐ ♣➦❡♥ést ♥❛ AQn s ✭✈í❝❡✮ ✈❛❞♥ý♠✐ ❤r❛♥❛♠✐✳ P♦❞♦❜♥ý♠ ③♣➲s♦❜❡♠ t❛❦é ❞♦❦á➸❡♠❡✱ ➸❡ ❥❡st❧✐➸❡ f ≤ 3n − 7 ❛ ❦❛➸❞ý ✈r❝❤♦❧ AQn s♦✉s❡❞í ❛s♣♦➡ s❡ ❞✈➙♠❛ ③❞r❛✈ý♠✐ ❤r❛♥❛♠✐✱ ♣❛❦ AQn ♦❜s❛❤✉❥❡ ❤❛♠✐❧t♦♥♦✈s❦♦✉ ❦r✉➸♥✐❝✐ t✈♦➦❡♥♦✉ ♣♦✉③❡ ③❞r❛✈ý♠✐ ❤r❛♥❛♠✐✳ ◆❛✈í❝ ❞♦❦á➸❡♠❡✱ ➸❡ ❦❛➸❞é ❞✈❛ ♠♦♥♦♠♦r✜s♠② G1 ❞♦ AQn ❛ G2 ❞♦ AQm ❧③❡ s❧♦➸✐t ♥❛ ♠♦♥♦♠♦r✜s♠✉s ❦❛rté③s❦é❤♦ s♦✉↔✐♥✉ G1 G2 ❞♦ AQn+m✳ ❑❧í↔♦✈á s❧♦✈❛✿ ❤②♣❡r❦r②❝❤❧❡✱ ❛✉❣♠❡♥t❡❞ ❝✉❜❡✱ ❤❛♠✐❧t♦♥♦✈s❦é ❦r✉➸♥✐❝❡✱ ✈❛❞♥é ❤r❛✲ ♥②

❚✐t❧❡✿ P❛rt✐t✐♦♥s ♦❢ ✐♥t❡r❝♦♥♥❡❝t✐♦♥ ♥❡t✇♦r❦s ✐♥t♦ ❧♦♥❣ ♣❛t❤s ❆✉t❤♦r✿ ❑r②➨t♦❢ ▼➙❦✉t❛ ❉❡♣❛rt♠❡♥t✿ ❉❡♣❛rt♠❡♥t ♦❢ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❛♥❞ ▼❛t❤❡♠❛t✐❝❛❧ ▲♦❣✐❝ ❙✉♣❡r✈✐s♦r✿ ▼❣r✳ P❡tr ●r❡❣♦r✱ P❤✳❉✳✱ ❑❚■▼▲ ❆❜str❛❝t✿ ❲❡ s✉r✈❡② s♦♠❡ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ s❡❧❡❝t❡❞ ✐♥t❡r❝♦♥♥❡❝t✐♦♥ ♥❡t✇♦r❦s✱ ♠♦r❡ ♣r❡❝✐s❡❧② ♠♦❞✐✜❝❛t✐♦♥s ♦❢ ❤②♣❡r❝✉❜❡s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ st✉❞② ❢❛✉❧t✲t♦❧❡r❛♥❝❡ ♦❢ ❛✉❣♠❡♥t❡❞ ❝✉❜❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❍❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s✳ ❲❡ ❛❧✲ s♦ ♣r❡s❡♥t ❛ ♠❡t❤♦❞ t♦ ❝♦♥str✉❝t ♠❛♥② s♣❛♥♥✐♥❣ s✉❜❣r❛♣❤s ♦❢ AQn ✐s♦♠♦r♣❤✐❝ t♦ Qn ✉s✐♥❣ ❡❧❡♠❡♥t❛r② ❢❛❝ts ❢r♦♠ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ❲❡ ♣r♦✈❡ t❤❛t n✲❞✐♠❡♥s✐♦♥❛❧ ❛✉❣♠❡♥t❡❞ ❝✉❜❡ AQn ✇✐t❤ f ❢❛✉❧t② ❡❞❣❡s ❝♦♥t❛✐♥s ❛ ❝♦♣② ♦❢ n✲❞✐♠❡♥s✐♦♥❛❧ ❤②♣❡r✲ ❝✉❜❡ Qn ✇✐t❤ ❛t ♠♦st n 2n−1 f ❢❛✉❧t② ❡❞❣❡s✳ ❯s✐♥❣ t❤✐s r❡s✉❧t ✇❡ ❛r❡ ❛❜❧❡ t♦ ❡❛s✐❧② tr❛♥s❢❡r s❡✈❡r❛❧ ♣r♦♣❡rt✐❡s ♦❢ Qn ✇✐t❤ ❢❛✉❧t② ❡❞❣❡s t♦ AQn ✇✐t❤ ✭♠♦r❡✮ ❢❛✉❧t② ❡❞❣❡s✳ ❆♣♣❧②✐♥❣ s✐♠✐❧❛r ♠❡t❤♦❞✱ ✇❡ ❛❧s♦ s❤♦✇ t❤❛t ✐❢ f ≤ 3n−7 ❛♥❞ ❡✈❡r② ✈❡rt❡① ♦❢ AQn ✐s ✐♥❝✐❞❡♥t t♦ ❛t ❧❡❛st t✇♦ ♥♦♥✲❢❛✉❧t② ❡❞❣❡s✱ t❤❡♥ AQn ❤❛s ❛ ♥♦♥✲❢❛✉❧t② ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡✳ ▼♦r❡♦✈❡r✱ ✇❡ s❤♦✇ t❤❛t ❡✈❡r② t✇♦ ♠♦♥♦♠♦r♣❤✐s♠s ♦❢ G1 t♦ AQn ❛♥❞ G2 t♦ AQm ❝❛♥ ❜❡ ❝♦♠♣♦s❡❞ ✐♥t♦ ❛ ♠♦♥♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t G1 G2 t♦ AQn+m✳ ❑❡②✇♦r❞s✿ ❤②♣❡r❝✉❜❡✱ ❛✉❣♠❡♥t❡❞ ❝✉❜❡✱ ❤❛♠✐❧t♦♥✐❝✐t②✱ ❢❛✉❧t② ❡❞❣❡s