Tema2. Fundamentos y Aplicaciones de La Teoria de Grafos. Diagramas en Árbol PDF

 

 

󰁔󰁥󰁭󰁡2 󰁆󰁵󰁮󰁤󰁡󰁭󰁥󰁮󰁴󰁯󰁳 󰁹 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰁯󰁮󰁥󰁳 󰁤󰁥 󰁬󰁡 󰁴󰁥󰁯󰁲󰁩󰁡 󰁤󰁥 󰁧󰁲󰁡󰁦󰁯󰁳. 󰁄󰁩󰁡󰁧󰁲󰁡󰁭󰁡󰁳 󰁥󰁮 󰃡󰁲󰁢󰁯󰁬

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

󰁉󰁮󰁤󰁩󰁣󰁥󰀺 1.I󰁮󰁴󰁲󰁯󰁤󰁵󰁣󰁣󰁩󰃳󰁮 󰁹 󰁤󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰁯󰁮󰁥󰁳 1.1.I󰁮󰁴󰁲󰁯󰁤󰁵󰁣󰁣󰁩󰃳󰁮 1.2. G󰁲󰁡󰁦󰁯 󰁳󰁩󰁭󰁰󰁬󰁥 1.3.M󰁵󰁬󰁴󰁩󰁧󰁲󰁡󰁦󰁯 󰁤󰁥 󰁯󰁲󰁤󰁥󰁮 󰁰 1.4.G󰁲󰁡󰁦󰁯 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯 1.5.R󰁥󰁤󰁥󰁳 1.6. O󰁲󰁤󰁥󰁮, 󰁴󰁡󰁭󰁡󰃱󰁯 󰁹 󰁧󰁲󰁡󰁤󰁯 󰁤󰁥 󰁩󰁮󰁣󰁩󰁤󰁥󰁮󰁣󰁩󰁡. 1.7 C󰁡󰁤󰁥󰁮󰁡󰁳 󰁹 󰁣󰁩󰁣󰁬󰁯󰁳 1.8 C󰁡󰁭󰁩󰁮󰁯󰁳 󰁹 󰁣󰁩󰁲󰁣󰁵󰁩󰁴󰁯󰁳 1.9 󰁳󰁵󰁢󰁧󰁲󰁡󰁦󰁯󰁳 2. G󰁲󰁡󰁦󰁯󰁳 󰁣󰁯󰁮󰁥󰁸󰁯󰁳 2.1 D󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮 󰁧󰁲󰁡󰁦󰁯󰁳 󰁣󰁯󰁮󰁥󰁸󰁯󰁳. 2.2 C󰁡󰁲󰁡󰁣󰁴󰁥󰁲󰁩󰁺󰁡󰁣󰁩󰃳󰁮 󰁤󰁥 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁣󰁯󰁮󰁥󰁸󰁯󰁳. G󰁲󰁡󰁦󰁯 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯 󰁦󰁵󰁥󰁲󰁴󰁥󰁭󰁥󰁮󰁴󰁥 󰁣󰁯󰁮󰁥󰁸󰁯 3. G󰁲󰁡󰁦󰁯 󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯, 󰁰󰁵󰁥󰁮󰁴󰁥󰁳 󰁤󰁥 󰁫󰃶󰁮󰁩󰁳󰁧󰁵󰁥󰁲. 3.1 E󰁬 󰁰󰁲󰁯󰁢󰁲󰁥󰁬󰁡 󰁤󰁥 󰁬󰁯󰁳 󰁰󰁵󰁥󰁮󰁴󰁥󰁳 󰁤󰁥 K󰁲󰃶󰁮󰁩󰁳󰁧󰁵󰁥󰁲 3.2 G󰁲󰁡󰁦󰁯 󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯 󰁹 󰁳󰁥󰁭󰁩󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯 4. 󰁧󰁲󰁡󰁦󰁯󰁳 󰁨󰁡󰁭󰁩󰁬󰁴󰁯󰁮󰁩󰁡󰁮󰁡󰁮󰁯 5 G󰁲󰁡󰁦󰁯󰁳 󰁰󰁬󰁡󰁮󰁯󰁳 6. D󰁩󰁡󰁧󰁲󰁡󰁭󰁡󰁳 󰁤󰁥 󰃡󰁲󰁢󰁯󰁬. 7. M󰁡󰁴󰁲󰁩󰁣󰁥󰁳 󰁡󰁳󰁯󰁣󰁩󰁡󰁤󰁡󰁳 󰁡 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯

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󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

 

1.󰀭󰁉󰁎󰁔󰁒󰁏󰁄󰁕󰁃󰁃󰁉󰃓󰁎 L󰁡 󰁴󰁥󰁯󰁲󰃭󰁡 󰁤󰁥 󰁧󰁲󰁡󰁦󰁯󰁳 󰁮󰁡󰁣󰁩󰃳 󰁣󰁯󰁭󰁯 󰁵󰁮󰁡 󰁲󰁡󰁭󰁡 󰁤󰁥 󰁬󰁡 T󰁯󰁰󰁯󰁬󰁯󰁧󰃭󰁡, 󰁹 󰁨󰁯󰁹 󰁥󰁮 󰁤󰃭󰁡 󰁳󰁥 󰁨󰁡 󰁣󰁯󰁮󰁶󰁥󰁲󰁴󰁩󰁤󰁯 󰁥󰁮 󰁵󰁮󰁡 󰁨󰁥󰁲󰁲󰁡󰁭󰁩󰁥󰁮󰁴󰁡 󰁭󰁡󰁴󰁥󰁭󰃡󰁴󰁩󰁣󰁡 󰁩󰁮󰁤󰁩󰁳󰁰󰁥󰁮󰁳󰁡󰁢󰁬󰁥 󰁥󰁮 󰁣󰁡󰁭󰁰󰁯󰁳 󰁭󰁵󰁹 󰁤󰁩󰁶󰁥󰁲󰁳󰁯󰁳. D󰁥 󰁦󰁯󰁲󰁭󰁡 󰁧󰁥󰁮󰁥󰁲󰁡󰁬 󰁰󰁵󰁥󰁤󰁥 󰁤󰁥󰁣󰁩󰁲󰁳󰁥 󰁱󰁵󰁥 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁳󰁥 󰁵󰁴󰁩󰁬󰁩󰁺󰁡󰁮 󰁰󰁡󰁲 󰁲󰁥󰁰󰁲󰁥󰁳󰁥󰁮󰁴󰁡󰁲 󰁬󰁡󰁳 󰁲󰁥󰁬󰁡󰁣󰁩󰁯󰁮󰁥󰁳 󰁥󰁮󰁴󰁲󰁥 󰁬󰁯󰁳 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁤󰁥 󰁵󰁮 󰁣󰁯󰁮󰁪󰁵󰁮󰁴󰁯. M󰁥󰁤󰁩󰁡󰁮󰁴󰁥 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁰󰁯󰁤󰁥󰁭󰁯󰁳 󰁲󰁥󰁳󰁯󰁬󰁶󰁥󰁲 󰁹 󰁰󰁬󰁡󰁮󰁴󰁥󰁡󰁲 󰁮󰁵󰁭󰁥󰁲󰁯󰁳󰁯󰁳 󰁰󰁲󰁯󰁢󰁬󰁥󰁭󰁡󰁳 󰁲󰁥󰁡󰁬󰁥󰁳 󰁤󰁥 󰁬󰁡 󰁶󰁩󰁤󰁡 󰁤󰁩󰁡󰁲󰁩󰁡, 󰁣󰁯󰁭󰁯 󰁰󰁵󰁥󰁤󰁥󰁮 󰁳󰁥󰁲 󰁤󰁥 󰁣󰁩󰁲󰁣󰁵󰁬󰁡󰁣󰁩󰃳󰁮 (󰁡󰃩󰁲󰁥󰁡, 󰁣󰁡󰁲󰁲󰁥󰁴󰁥󰁲󰁡󰁳, 󰁤󰁥 󰁧󰁡󰁳...) 󰁯 󰁤󰁥 󰁲󰁥󰁤󰁥󰁳 (󰁩󰁮󰁦󰁯󰁲󰁭󰃡󰁴󰁩󰁣󰁡󰁳, 󰁥󰁬󰃩󰁣󰁴󰁲󰁩󰁣󰁡󰁳...). L󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁳󰁯󰁮 󰁵󰁳󰁡󰁤󰁯󰁳 󰁥󰁮 󰁣󰁡󰁭󰁰󰁯󰁳 󰁴󰁡󰁮 󰁤󰁩󰁳󰁰󰁡󰁲󰁥󰁳 󰁣󰁯󰁭󰁯 󰁰󰁵󰁥󰁤󰁥󰁮 󰁳󰁥󰁲 󰁬󰁡 Q󰁵󰃭󰁭󰁩󰁣󰁡, G󰁥󰁮󰃩󰁴󰁩󰁣󰁡, I󰁮󰁦󰁯󰁲󰁭󰃡󰁴󰁩󰁣󰁡, T󰁥󰁯󰁲󰃭󰁡 󰁤󰁥 S󰁩󰁳󰁴󰁥󰁭󰁡󰁳.

2. 󰁄󰁅󰁆󰁉󰁎󰁉󰁃󰁉󰁏󰁎󰁅󰁓 1.1.GRAFO SIMPLE  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮󰀱󰀺 U󰁮  󰁧󰁲󰁡󰁦󰁯 󰁳󰁩󰁭󰁰󰁬󰁥  󰁥󰁳 󰁵󰁮 󰁰󰁡󰁲 (V,E), 󰁤󰁯󰁮󰁤󰁥 V 󰁥󰁳 󰁵󰁮 󰁣󰁯󰁮󰁪󰁵󰁮󰁴󰁯 󰁤󰁥

󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳, 󰁹 E 󰁥󰁳 󰁧󰁲󰁵󰁰󰁯 󰁤󰁥 󰁰󰁡󰁲󰁥󰁳 󰁤󰁥 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁤󰁥 V, 󰁤󰁯󰁮󰁤󰁥 󰁮󰁯 󰁳󰁥 󰁰󰁥󰁲󰁭󰁩󰁴󰁥 󰁬󰁡 󰁲󰁥󰁰󰁥󰁴󰁩󰁣󰁩󰃳󰁮 󰁤󰁥 󰁮󰁩󰁮󰁧󰁵󰁮󰁯 󰁹 󰁮󰁯 󰁩󰁭󰁰󰁯󰁲󰁴󰁡 󰁬󰁡 󰁯󰁲󰁤󰁥󰁮󰁡󰁣󰁩󰃳󰁮 󰁤󰁥󰁬 󰁰󰁡󰁲 󰁤󰁥 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳. E=󰁻󰁥 󰁪=󰁥 󰁪1,󰁥 󰁪2: 󰁥󰁪1, 󰁥󰁪2∈V󰁽  󰁅󰁪󰁥󰁭󰁰󰁬󰁯󰀺 V=󰁻1,2,3󰁽 E=󰁻󰁻1,2󰁽,󰁻2,3󰁽󰁽

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P󰁵󰁥󰁤󰁥 󰁲󰁥󰁰󰁲󰁥󰁳󰁥󰁮󰁴󰁡󰁲 󰁬󰁡󰁳 󰁣󰁯󰁮󰁥󰁸󰁩󰁯󰁮󰁥󰁳 󰁡󰃩󰁲󰁥󰁡󰁳 󰁥󰁮󰁴󰁲󰁥 3 󰁣󰁩󰁵󰁤󰁡󰁤󰁥󰁳, 󰁯󰁲󰁤󰁥󰁮󰁡󰁤󰁯󰁲󰁥󰁳 󰁣󰁯󰁮󰁥󰁣󰁴󰁡󰁤󰁯󰁳 󰁰󰁯󰁲 󰁲󰁥󰁤, 󰁯 󰁬󰁯󰁳 󰁮󰃺󰁭󰁥󰁲󰁯󰁳 󰁤󰁥󰁬 1 󰁡󰁬 3 󰁣󰁵󰁹󰁡󰁳 󰁤󰁩󰁦󰁥󰁲󰁥󰁮󰁣󰁩󰁡󰁳 󰁴󰁥󰁮󰁧󰁡󰁮 󰁭󰁯󰁤󰁵󰁬󰁯 1. 3 E󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁤󰁥 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁳󰁩󰁭󰁰󰁬󰁥󰁳: •  󰁕󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁯 󰁮󰁯󰁤󰁯 󰁳󰁯󰁮 󰁣󰁡󰁤󰁡 󰁵󰁮󰁯 󰁤󰁥 󰁬󰁯󰁳 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁤󰁥 V •   󰁅󰁪󰁥󰁳 󰁯 󰁡󰁲󰁩󰁳󰁴󰁡󰁳 󰁳󰁯󰁮 󰁣󰁡󰁤󰁡 󰁵󰁮󰁡 󰁤󰁥 󰁬󰁡󰁳 󰁵󰁮󰁩󰁯󰁮󰁥󰁳 󰁥󰁮󰁴󰁲󰁥 󰁤󰁯󰁳 󰁰󰁡󰁲󰁥󰁳 󰁤󰁥 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁤󰁥 V(󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳); 󰁳󰁯󰁮 󰁬󰁡󰁳 󰁰󰁡󰁲󰁥󰁪󰁡󰁳 󰁤󰁥 V 󰁱󰁵󰁥 󰁦󰁯󰁲󰁭󰁡󰁮 E. •  󰁖󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁡󰁤󰁹󰁡󰁣󰁥󰁮󰁴󰁥󰁳: D󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩, 󰁪 ∈V 󰁳󰁯󰁮 󰁡󰁤󰁹󰁡󰁣󰁥󰁮󰁴󰁥󰁳 󰁳󰁩 󰁥󰁳󰁴󰃡󰁮 󰁣󰁯󰁮󰁥󰁣󰁴󰁡󰁤󰁯󰁳 󰁰󰁯󰁲 󰁡󰁬󰁧󰃺󰁮 󰁥󰁪󰁥   󰁻󰁩,󰁪󰁽 󰃳 󰁻󰁪,󰁩󰁽 ∈E •   󰁂󰁵󰁣󰁬󰁥󰀺 C󰁵󰁡󰁮󰁤󰁯 󰁥󰁮 󰁵󰁮 󰁥󰁪󰁥 󰁬󰁯󰁳 󰁥󰁸󰁴󰁲󰁥󰁭󰁯󰁳 󰁣󰁯󰁩󰁮󰁣󰁩󰁤󰁥󰁮, 󰁻󰁩,󰁩󰁽. T󰁩󰁰󰁯󰁳 󰁤󰁥 󰁧󰁲󰁡󰁦󰁯󰁳: •  G󰁲󰁡󰁦󰁯 󰁳󰁩󰁭󰁰󰁬󰁥: 󰁣󰁵󰁡󰁮󰁤󰁯 󰁮󰁯 󰁣󰁯󰁮󰁴󰁩󰁥󰁮󰁥 󰁮󰁩󰁮󰁧󰃺󰁮 󰁢󰁵󰁣󰁬󰁥. •  G󰁲󰁡󰁦󰁯 󰁮󰁵󰁬󰁯: 󰁣󰁵󰁡󰁮󰁤󰁯 󰁮󰁯 󰁣󰁯󰁮󰁴󰁩󰁥󰁮󰁥 󰁥󰁪󰁥󰁳, E=∅  •  G󰁲󰁡󰁦󰁯 󰁣󰁯󰁭󰁰󰁬󰁥󰁴󰁯 󰁣󰁵󰁡󰁮󰁤󰁯 󰁴󰁩󰁥󰁮󰁥 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁥󰁪󰁥󰁳 󰁰󰁯󰁳󰁩󰁢󰁬󰁥󰁳, 󰁩󰁮󰁣󰁬󰁵󰁹󰁥󰁮󰁤󰁯 󰁬󰁯󰁳  󰁢󰁵󰁣󰁬󰁥󰁳. •  G󰁲󰁡󰁦󰁯 󰁳󰁩󰁭󰁰󰁬󰁥󰁭󰁥󰁮󰁴󰁥 󰁣󰁯󰁭󰁰󰁬󰁥󰁴󰁯 󰁣󰁵󰁡󰁮󰁤󰁯 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁳󰁯󰁮 󰁡󰁤󰁹󰁡󰁣󰁥󰁮󰁴󰁥󰁳, 󰁥󰁬 󰁧󰁲󰁡󰁦󰁯 󰁴󰁩󰁥󰁮󰁥 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁥󰁪󰁥󰁳 󰁥󰁸󰁣󰁥󰁰󰁴󰁯 󰁬󰁯󰁳 󰁢󰁵󰁣󰁬󰁥󰁳

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󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

  G󰁲󰁡󰁦󰁯 N󰁵󰁬󰁯 G󰁲󰁡󰁦󰁯 󰁣󰁯󰁭󰁰󰁬󰁥󰁴󰁯

G󰁲󰁡󰁦󰁯 󰁳󰁩󰁭󰁰󰁬󰁥󰁭󰁥󰁮󰁴󰁥 󰁣󰁯󰁭󰁰󰁬󰁥󰁴󰁯 1.2.MULTIGRAFO DE ORDEN P  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮󰀺 󰁕󰁮 󰁰󰀭󰁧󰁲󰁡󰁦󰁯 󰁥󰁳 󰁵󰁮 󰁰󰁡󰁲 (V,E) 󰁤󰁯󰁮󰁤󰁥 V 󰁥󰁳 󰁵󰁮 󰁣󰁯󰁮󰁪󰁵󰁮󰁴󰁯 󰁤󰁥 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁹 E 󰁥󰁳

󰁵󰁮 󰁣󰁯󰁮󰁪󰁵󰁮󰁴󰁯 󰁤󰁥 󰁰󰁡󰁲󰁥󰁳 󰁤󰁥 V, 󰁤󰁯󰁮󰁤󰁥 󰁥󰁬 󰁯󰁲󰁤󰁥󰁮 󰁤󰁥 󰁬󰁯󰁳 󰁰󰁡󰁲󰁥󰁪󰁡󰁳 󰁮󰁯 󰁩󰁭󰁰󰁯󰁲󰁴󰁡, 󰁹 󰁤󰁯󰁮󰁤󰁥 󰁵󰁮󰁡 󰁰󰁡󰁲󰁥󰁪󰁡  󰁰󰁵󰁥󰁤󰁥 󰁡󰁰󰁡󰁲󰁥󰁣󰁥󰁲 󰁵󰁮 󰁭󰃡󰁸󰁩󰁭󰁯 󰁤󰁥 󰁰 󰁶󰁥󰁣󰁥󰁳 󰁶 󰁥󰁣󰁥󰁳 󰁥󰁮 E. E=󰁻󰁥 󰁪=󰁻󰁥 󰁪1,󰁥 󰁪2󰁽, 󰁥 󰁪1, 󰁥 󰁪2∈E : I(󰁥 󰁪)=∑󰁶󰁥󰁣󰁥󰁳󰁪1≤ 󰁰 ∀ 󰁪󰁽 C󰁯󰁮 󰁥󰁳󰁴󰁡 󰁤󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁳󰁩󰁭󰁰󰁬󰁥 󰁥󰁳 󰁵󰁮 1󰀭󰁧󰁲󰁡󰁦󰁯. L󰁯󰁳 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁤󰁥 󰁬󰁯󰁳 󰁰󰀭󰁧󰁲󰁡󰁦󰁯󰁳 󰁳󰁯󰁮 󰁬󰁯󰁳 󰁭󰁩󰁳󰁭󰁯󰁳 󰁤󰁥 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁳󰁩󰁭󰁰󰁬󰁥󰁳. U󰁮 󰁥󰁪󰁥󰁭󰁰󰁬󰁯 󰁣󰁬󰃡󰁳󰁩󰁣󰁯 󰁥󰁳 󰁥󰁬 󰁭󰁵󰁬󰁴󰁩󰁧󰁲󰁡󰁦󰁯 󰁤󰁥 󰁯󰁲󰁤󰁥󰁮 2 󰁱󰁵󰁥 󰁰󰁲󰁥󰁴󰁥󰁮󰁤󰁥 󰁥󰁳󰁴󰁵󰁤󰁩󰁡󰁲 󰁬󰁯󰁳 7 󰁰󰁵󰁥󰁮󰁴󰁥󰁳 󰁤󰁥 K󰃶󰁮󰁩󰁧󰁳󰁢󰁥󰁲󰁧(P󰁲󰁵󰁳󰁩󰁡) 󰁱󰁵󰁥 󰁵󰁮󰁥 󰁬󰁡󰁳 󰁯󰁲󰁩󰁬󰁬󰁡󰁳 A,C 󰁹 󰁤󰁯󰁳 󰁩󰁳󰁬󰁡󰁳 B,D: 󰁥6

󰁂  󰁥󰀳

󰁥󰀲

󰁁 

󰁖󰀽󰁻󰁁󰀬󰁂󰀬󰁃󰀬󰁄󰁽 󰁖󰀽󰁻󰁁󰀬󰁂󰀬󰁃󰀬󰁄󰁽

󰁃 

󰁅󰀽󰁻󰀨󰁁󰀬󰁂󰀩󰀬󰀨󰁁󰀬󰁂󰀩󰀬󰀨󰁁󰀬󰁄󰀩󰀬󰀨󰁁 󰁅󰀽󰁻󰀨󰁁󰀬󰁂󰀩󰀬󰀨󰁁󰀬 󰁂󰀩󰀬󰀨󰁁󰀬󰁄󰀩󰀬󰀨󰁁󰀬󰁄󰀩󰀬󰀨󰁁󰀬󰁃󰀩󰀬󰀨󰁂 󰀬󰁄󰀩󰀬󰀨󰁁󰀬󰁃󰀩󰀬󰀨󰁂󰀬󰁃󰀩󰀬󰀨󰁄󰀬󰁃󰀩󰁽 󰀬󰁃󰀩󰀬󰀨󰁄󰀬󰁃󰀩󰁽 

󰁥󰀱 󰁥󰀴

󰁥󰀵

󰁥󰀷

󰁄

L󰁡 󰁭󰁡󰁹󰁯󰁲󰃭󰁡 󰁤󰁥 󰁡󰁵󰁴󰁯󰁲󰁥󰁳 󰁣󰁯󰁩󰁮󰁣󰁩󰁤󰁥󰁮 󰁥󰁮 󰁣󰁯󰁮󰁳󰁩󰁤󰁥󰁲󰁡󰁲 󰁥󰁬 󰁮󰁡󰁣󰁩󰁭󰁩󰁥󰁮󰁴󰁯 󰁤󰁥󰁬 󰁮󰁡󰁣󰁩󰁭󰁩󰁥󰁮󰁴󰁯 󰁤󰁥󰁬 󰁧󰁲󰁡󰁦󰁯 󰁣󰁵󰁡󰁮󰁤󰁯 E󰁵󰁬󰁥󰁲(S XVIII) 󰁥󰁳󰁴󰁵󰁤󰁩󰁡󰁢󰁡 󰁥󰁳󰁴󰁥 2󰀭󰁧󰁲󰁡󰁦󰁯 󰁣󰁯󰁭󰁯 󰁰󰁡󰁳󰁡󰁴󰁩󰁥󰁭󰁰󰁯, 󰁶󰁩󰁥󰁮󰁤󰁯 󰁬󰁡 󰁦󰁯󰁲󰁭󰁡 󰁤󰁥 󰁣󰁲󰁵󰁺󰁡󰁲 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁰󰁵󰁥󰁮󰁴󰁥󰁳 󰁳󰁩󰁮 󰁰󰁡󰁳󰁡󰁲 󰁤󰁯󰁳 󰁶󰁥󰁣󰁥󰁳 󰁰󰁯󰁲 󰁥󰁬 󰁭󰁩󰁳󰁭󰁯.

4

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

1.3.GRAFO DIRIGIDO U ORIENTADO A 󰁶󰁥󰁣󰁥󰁳 󰁬󰁡󰁳 󰁲󰁥󰁬󰁡󰁣󰁩󰁯󰁮󰁥󰁳 󰁥󰁮󰁴󰁲󰁥 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁮󰁯 󰁳󰁯󰁮 󰁳󰁩󰁭󰃩󰁴󰁲󰁩󰁣󰁡󰁳 󰁹 󰁮󰁯 󰁥󰁳 󰁥󰁱󰁵󰁩󰁶󰁡󰁬󰁥󰁮󰁴󰁥 󰁻󰁥󰁩,󰁥󰁪󰁽 󰁡 󰁻󰁥󰁪,󰁥󰁩󰁽 󰁣󰁯󰁭󰁯 󰁥󰁲󰁡 󰁥󰁮 󰁥󰁬 󰁣󰁡󰁳󰁯 󰁤󰁥 󰁬󰁯󰁳 󰁰󰀭󰁧󰁲󰁡󰁦󰁯󰁳. P󰁯󰁲 󰁥󰁪󰁥󰁭󰁰󰁬󰁯 󰁬󰁡 󰁲󰁥󰁬󰁡󰁣󰁩󰃳󰁮 󰁤󰁥 󲀜󰁨󰁩󰁪󰁯󰁳 󰁤󰁥󲀝, 󲀜󰁤󰁩󰁶󰁩󰁳󰁯󰁲󰁥󰁳 󰁤󰁥󲀝 󰁮󰁯 󰁴󰁩󰁥󰁮󰁥󰁮 󰁥󰁳󰁡 󰁢󰁩󰁤󰁩󰁲󰁥󰁣󰁣󰁩󰁯󰁮󰁡󰁬󰁩󰁤󰁡󰁤; 󰁳󰁵󰁲󰁧󰁥 󰁡󰁳󰃭 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯󰁳.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮󰀺  󰁇󰁲󰁡󰁦󰁯 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯 󰁥󰁳 󰁵󰁮 󰁰󰁡󰁲 (V,A) 󰁤󰁯󰁮󰁤󰁥 V 󰁥󰁳 󰁵󰁮 󰁣󰁯󰁮󰁪󰁵󰁮󰁴󰁯 󰁤󰁥 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳

󰁹 A 󰁥󰁳 󰁵󰁮 󰁣󰁯󰁮󰁪󰁵󰁮󰁴󰁯 󰁦󰁯󰁲󰁭󰁡󰁤󰁯 󰁰󰁯󰁲 󰁰󰁡󰁲󰁥󰁪󰁡󰁳 󰁤󰁥 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁤󰁥 V, 󰁤󰁯󰁮󰁤󰁥 󰁳󰁩 󰁩󰁭󰁰󰁯󰁲󰁴󰁡 󰁥󰁬 󰁯󰁲󰁤󰁥󰁮 󰁤󰁥 󰁬󰁯󰁳 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁤󰁥󰁬 󰁰󰁡󰁲, 󰁳󰁩󰁥󰁮󰁤󰁯 󰁤󰁩󰁳󰁴󰁩󰁮󰁴󰁯 󰁻󰁥󰁩,󰁥󰁪󰁽 󰁡 󰁻󰁥󰁪,󰁥󰁩󰁽󰁣󰁯󰁮 󰁥󰁪 󰁹 󰁥󰁩 󰁰󰁥󰁲󰁴󰁥󰁮󰁥󰁣󰁩󰁥󰁮󰁴󰁥󰁳 󰁡 V. V.  󰁅󰁪󰁥󰁭󰁰󰁬󰁯: V=󰁻󰁡,󰁢,󰁣,󰁤󰁽 A=󰁻(󰁡,󰁢),(󰁢,󰁡),(󰁤,󰁢),(󰁣,󰁢)󰁽.

E󰁮 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯󰁳 󰁥󰁬 󰁥󰁪󰁥 󰁳󰁥 󰁬󰁬󰁡󰁭󰁡 󰁡󰁲󰁣󰁯.  󰁁󰁲󰁣󰁯 󰁥󰁳 󰁣󰁡󰁤󰁡 󰁰󰁡󰁲 󰁤󰁥 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁱󰁵󰁥 󰁦󰁯󰁲󰁭󰁡 󰁰󰁡󰁲󰁴󰁥 󰁤󰁥 A. L󰁯󰁳 󰁤󰁥󰁭󰃡󰁳 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁳󰁯󰁮 󰁬󰁯󰁳 󰁭󰁩󰁳󰁭󰁯󰁳 󰁱󰁵󰁥 󰁥󰁮 󰁧󰁲󰁡󰁦󰁯󰁳 󰁮󰁯 󰁯󰁲󰁩󰁥󰁢󰁴󰁡󰁤󰁯󰁳 T󰁯󰁤󰁯 󰁧󰁲󰁡󰁦󰁯 󰁮󰁯 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯 󰁳󰁥 󰁰󰁵󰁥󰁤󰁥 󰁣󰁯󰁮󰁳󰁩󰁤󰁥󰁲󰁡󰁲 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯 󰁳󰁩 󰁡 󰁣󰁡󰁤󰁡 󰁥󰁪󰁥 󰁻󰁩,󰁪󰁽 󰁬󰁯 󰁳󰁵󰁳󰁴󰁩󰁴󰁵󰁩󰁭󰁯󰁳 󰁰󰁯󰁲 󰁤󰁯󰁳 󰁡󰁲󰁣󰁯󰁳 󰁻󰁩,󰁪󰁽 󰁹 󰁻󰁪,󰁩󰁽. 󰁩 󰁪 󰁩 󰁪 D󰁡󰁤󰁯 󰁵󰁮 󰁡󰁲󰁣󰁯 󰁻󰁩,󰁪󰁽, 󰁩 󰁳󰁥󰁲󰃡 󰁥󰁬 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁩󰁮󰁩󰁣󰁩󰁡󰁬 󰁹 󰁪 󰁥󰁬 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁦󰁩󰁮󰁡󰁬. S󰁥 󰁤󰁩󰁣󰁥 󰁱󰁵󰁥 󰁩 󰁥󰁳 

 󰁰󰁲󰁥󰁤󰁥󰁣󰁥󰁳󰁯󰁲 󰁤󰁥 󰁪 󰁹 󰁪 󰁳󰁵󰁣󰁥󰁳󰁯󰁲 󰁳󰁵󰁣󰁥󰁳 󰁯󰁲 󰁤󰁥 󰁩. 󰁩  󰁪 1.4. OPERACIONES Y APLICACIONES EN LOS GRAFOS. 1.4.1. I󰁳󰁯󰁭󰁯󰁲󰁦󰁩󰁳󰁭󰁯 M󰁵󰁣󰁨󰁡󰁳 󰁶󰁥󰁣󰁥󰁳 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 (󰁯 󰁰󰀭󰁧󰁲󰁡󰁦󰁯󰁳) 󰁤󰁥 󰁡󰁰󰁡󰁲󰁩󰁥󰁮󰁣󰁩󰁡 󰁤󰁩󰁦󰁥󰁲󰁥󰁮󰁴󰁥, 󰁰󰁥󰁲󰁯 󰁱󰁵󰁥 󰁮󰁯󰁳 󰁤󰁡󰁮 󰁬󰁡 󰁭󰁩󰁳󰁭󰁡 󰁩󰁮󰁦󰁯󰁲󰁭󰁡󰁣󰁩󰃳󰁮. L󰁡 󰁤󰁩󰁦󰁥󰁲󰁥󰁮󰁣󰁩󰁡 󰁥󰁮󰁴󰁲󰁥 󰁥󰁳󰁴󰁯󰁳 󰁲󰁥󰁳󰁩󰁤󰁥 󰁥󰁮 󰁬󰁡 󰁤󰁩󰁳󰁴󰁩󰁮󰁴󰁡 󰁮󰁵󰁭󰁥󰁲󰁡󰁣󰁩󰃳󰁮 󰁤󰁥 󰁥󰁪󰁥󰁳 󰁹/󰁯 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳. C󰁵󰁡󰁮󰁤󰁯 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 󰁤󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁣󰁯󰁮 󰁭󰁩󰁳󰁭󰁡 󰁩󰁮󰁦󰁯󰁲󰁭󰁡󰁣󰁩󰃳󰁮 󰁤󰁥󰁣󰁩󰁭󰁯󰁳 󰁱󰁵󰁥 󰁥󰁳󰁴󰁯󰁳 󰁳󰁯󰁮 󰁩󰁳󰁯󰁭󰃳󰁲󰁦󰁩󰁣󰁯󰁳. V󰁥󰁡󰁭󰁯󰁳 󰁣󰁯󰁭󰁯 󰁰󰁯󰁤󰁥󰁭󰁯󰁳 󰁤󰁥󰁴󰁥󰁲󰁭󰁩󰁮󰁡󰁲 󰁳󰁩 󰁤󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁳󰁯 󰁩󰁳󰁯󰁭󰃳󰁲󰁦󰁩󰁣󰁯󰁳:  󰁉󰁳󰁯󰁭󰁯󰁲󰁦󰁩󰁳󰁭󰁯  󰁉󰁳󰁯󰁭󰁯󰁲󰁦󰁩󰁳 󰁭󰁯 󰁤󰁥 󰁧󰁲󰁡󰁦󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳: 󰁤󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁳󰁯󰁮 󰁩󰁳󰁯󰁭󰃳󰁲󰁦󰁩󰁣󰁯󰁳 󰁳󰁩 󰁥󰁸󰁩󰁳󰁴󰁥 󰁡󰁬󰁧󰁵󰁮󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮

Φ:V  V󰂴 󰁱󰁵󰁥 󰁣󰁵󰁭󰁰󰁬󰁡 󰁱󰁵󰁥 : S󰁩 󰁻󰁥󰁩,󰁥󰁪󰁽∈E   󰁻Φ(󰁥󰁩),Φ(󰁥󰁪)󰁽∈E󰂴

1.4.2 A󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰁯󰁮󰁥󰁳 A󰁤󰁹󰁡󰁣󰁥󰁮󰁴󰁥, A󰁮󰁴󰁥󰁣󰁥󰁳󰁯󰁲 󰁹 P󰁲󰁥󰁤󰁥󰁣󰁥󰁳󰁯󰁲. L󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁡󰁤󰁹󰁡󰁣󰁥󰁮󰁴󰁥 Γ   󰁵󰁮󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁤󰁥 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 (󰁰󰀭󰁧󰁲󰁡󰁦󰁯) 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯󰁳 󰁯 󰁮󰁯. Γ 󰁥󰁳    E󰁳󰁴󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁮󰁯󰁳 󰁲󰁥󰁬󰁡󰁣󰁩󰁯󰁮󰁡 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁱󰁵󰁥 󰁥󰁳󰁴󰃡󰁮 󰁵󰁮󰁩󰁤󰁯󰁳 󰁡 󰁯󰁴󰁲󰁯 󰁤󰁡󰁤󰁯 󰁭󰁥󰁤󰁩󰁡󰁮󰁴󰁥 󰁵󰁮 󰁥󰁪󰁥 (󰁡󰁲󰁣󰁯 󰁳󰁩 󰁥󰁳 󰁧󰁲󰁡󰁦󰁯 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯). Γ: V 󰁥󰁩 (󰁥󰁩,󰁥󰁪󰁽εA)

V 󰁥󰁪

: Γ(󰁥󰁩)=󰁻󰁥󰁪 : (󰁥󰁪,󰁥󰁩󰁽󰃳 (󰁥󰁩,󰁥󰁪)∈E󰁽 (E󰁮 󰁬󰁯󰁳 󰁰󰀭󰁧󰁲󰁡󰁦󰁯󰁳 󰁮󰁯󰁳 󰁴󰁩󰁥󰁮󰁥󰁮 󰁱󰁵󰁥 󰁤󰁥󰁣󰁩󰁲 󰁬󰁡󰁳 󰁶󰁥󰁣󰁥󰁳 󰁱󰁵󰁥

5

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

  A󰁳󰃭 󰁣󰁯󰁭󰁯 󰁬󰁡 󰁡󰁮󰁴󰁥󰁲󰁩󰁯󰁲 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁶󰃡󰁬󰁩󰁤󰁡 󰁴󰁡󰁮󰁴󰁯 󰁰󰁡󰁲󰁡 󰁧󰁲󰁡󰁦󰁯󰁳 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯󰁳 󰁣󰁯󰁭󰁯 󰁮󰁯, 󰁬󰁡󰁳 󰁤󰁯󰁳 󰁳󰁩󰁧󰁵󰁩󰁥󰁮󰁴󰁥󰁳 󰁳󰃳󰁬󰁯 󰁤󰁥󰁦󰁩󰁮󰁩󰁤󰁡󰁳 󰁥󰁮 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯󰁳: L󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁳󰁵󰁣󰁥󰁳󰁯󰁲   ,Γ󰁳, 󰁤󰁥 󰁵󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁤󰁥 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 (󰁰󰀭󰁧󰁲󰁡󰁦󰁯) 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯, 󰁥󰁳 󰁵󰁮󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁱󰁵󰁥 󰁮󰁯󰁳 󰁬󰁬󰁥󰁶󰁡 󰁡󰁬 󰁣󰁯󰁮󰁪󰁵󰁮󰁴󰁯 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁱󰁵󰁥 󰁥󰁳󰁴󰃡󰁮 󰁵󰁮󰁩󰁤󰁯󰁳 󰁣󰁯󰁮 󰁥󰁬 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁤󰁡󰁤󰁯 󰁭󰁥󰁤󰁩󰁡󰁮󰁴󰁥 󰁱󰁵󰁥: Γ󰁳󰁡󰁬󰁥 󰁤󰁥 󰃩󰁬. : (󰁥󰁩,󰁥󰁪)∈A󰁽 Γ󰁳󰁵󰁮  V :V󰁡󰁲󰁣󰁯 󰁳(󰁥󰁩)=󰁻󰁥󰁪, L󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁰󰁲󰁥󰁤󰁥󰁣󰁥󰁳󰁯󰁲   Γ 󰁰, 󰁥󰁳 󰁩󰁧󰁵󰁡󰁬 󰁱󰁵󰁥 󰁬󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁳󰁵󰁣󰁥󰁳󰁯󰁲, 󰁣󰁯󰁮 󰁬󰁡 󰁤󰁩󰁦󰁥󰁲󰁥󰁮󰁣󰁩󰁡 󰁱󰁵󰁥 󰁡󰁨󰁯󰁲󰁡 󰁬󰁯󰁳 󰁡󰁲󰁣󰁯󰁳 󰁬󰁬󰁥󰁧󰁡󰁮 󰁡󰁬 󰁶󰃩󰁲󰁴󰁩󰁣󰁥, 󰁹 󰁮󰁯 󰁳󰁡󰁬󰁥󰁮 󰁤󰁥 󰃩󰁬: Γ 󰁰:V  V : Γ 󰁰(󰁥󰁪)=󰁻󰁥󰁩 : (󰁥󰁩,󰁥󰁪)∈A󰁽 P󰁡󰁲󰁡 󰁣󰁯󰁮󰁯󰁣󰁥󰁲 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯 󰁢󰁡󰁳󰁴󰁡 󰁣󰁯󰁮 󰁣󰁯󰁮󰁯󰁣󰁥󰁲 V 󰁹 󰁬󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 Γ󰁳 󰁯 Γ 󰁰. 1.5. REDES P󰁯󰁤󰁥󰁭󰁯󰁳 󰁥󰁳󰁴󰁡󰁢󰁬󰁥󰁣󰁥󰁲 󰁱󰁵󰁥 󰁡 󰁣󰁡󰁤󰁡 󰁥󰁪󰁥 󰁯 󰁡󰁲󰁣󰁯 󰁬󰁥 󰁰󰁵󰁥󰁤󰁡 󰁡󰁳󰁩󰁧󰁮󰁡󰁲 󰁵󰁮 󰁶󰁡󰁬󰁯󰁲 󰁮󰁵󰁭󰃩󰁲󰁩󰁣󰁯 (󰁡󰁳󰁯󰁣󰁩󰁡󰁲󰁬󰁥 󰁵󰁮󰁡 󰁭󰁥󰁤󰁩󰁤󰁡) 󰁱󰁵󰁥 󰁮󰁯󰁳 󰁩󰁮󰁤󰁩󰁱󰁵󰁥 󰁬󰁡 󰁬󰁯󰁮󰁧󰁩󰁴󰁵󰁤, 󰁴󰁥󰁮󰁳󰁩󰃳󰁮 󰁥󰁮󰁴󰁲󰁥 󰁬󰁯󰁳 󰁮󰁵󰁤󰁯󰁳... E󰁳󰁴󰁯 󰁮󰁯󰁳 󰁤󰁡 󰁬󰁵󰁧󰁡󰁲 󰁡󰁬 󰁣󰁯󰁮󰁣󰁥󰁰󰁴󰁯 󰁤󰁥 󰁲󰁥󰁤󰁥󰁳 󰁯 󰁲󰁥󰁤󰁥󰁳 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁡󰁳.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: U󰁮󰁡 󰁲󰁥󰁤(󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁡󰀩  󰁥󰁳 󰁵󰁮󰁡 󰁴󰁥󰁲󰁮󰁡 (V,E,󰁬) 󰁤󰁯󰁮󰁤󰁥 (V,E) 󰁥󰁳 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯(󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯)

󰁹 󰁬󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁬:Eℜ󰁫 󰁱󰁵󰁥 󰁡󰁳󰁯󰁣󰁩󰁡 󰁡 󰁣󰁡󰁤󰁡 󰁥󰁪󰁥(󰁡󰁲󰁣󰁯) 󰁵󰁮 󰁶󰁡󰁬󰁯󰁲 󰁮󰁵󰁭󰃩󰁲󰁩󰁣󰁯, 󰁰󰁵󰁤󰁩󰁥󰁮󰁤󰁯 󰁳󰁥󰁲 󰁭󰁵󰁬󰁴󰁩󰁶󰁡󰁬󰁵󰁡󰁤󰁯.  󰁅󰁪󰁥󰁭󰁰󰁬󰁯: (V,E,󰁬)

V=󰁻L󰁥󰃳󰁮, S󰁡󰁬󰁡󰁭󰁡󰁮󰁣󰁡, V󰁡󰁬󰁬󰁡󰁤󰁯󰁬󰁩󰁤󰁽; E=󰁻(L󰁥󰃳󰁮, S󰁡󰁬󰁡󰁭󰁡󰁮󰁣󰁡),(S󰁡󰁬󰁡󰁭󰁡󰁮󰁣󰁡,V󰁡󰁬󰁬󰁡󰁤󰁯󰁬󰁩󰁤), (L󰁥󰃳󰁮,V󰁡󰁬󰁬󰁡󰁤󰁯󰁬󰁩󰁤)󰁽 󰁬= 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁱󰁵󰁥 󰁮󰁯󰁳 󰁤󰁡 󰁬󰁡 󰁤󰁩󰁳󰁴󰁡󰁮󰁣󰁩󰁡 󰁥󰁮 K󰁭 󰁹 󰁥󰁬 󰁴󰁩󰁥󰁭󰁰󰁯 󰁭󰁥󰁤󰁩󰁯 󰁤󰁥 󰁣󰁡󰁭󰁩󰁮󰁯  󰁰󰁯󰁲 󰁵󰁮 󰁡󰁵󰁴󰁯󰁢󰃺󰁳 󰁥󰁮󰁴󰁲󰁥 󰁬󰁡󰁳 3 󰁣󰁩󰁵󰁤󰁡󰁤󰁥󰁳 󰁥󰁮 󰁥 󰁮 󰁨󰁯󰁲󰁡󰁳: 󰁬(L󰁥󰃳󰁮,V󰁡󰁬󰁬󰁡󰁤󰁯󰁬󰁩󰁤)=170󰁫󰁭,1,8󰁨; 󰁬(L󰁥󰃳󰁮,V󰁡󰁬󰁬󰁡󰁤󰁯󰁬󰁩󰁤)=200󰁫󰁭,2,2󰁨 󰁬(V󰁡󰁬󰁬󰁡󰁤󰁯󰁬󰁩󰁤, S󰁡󰁬󰁡󰁭󰁡󰁮󰁣󰁡)=140󰁫󰁭,1,1󰁨.

1.6.ORDEN, TAMA󰃑O Y GRADO DE INCIDENCIA.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁏󰁲󰁤󰁥󰁮  󰁤󰁥 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯(󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯) G=(V,E) 󰁥󰁳 󰁥󰁬 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁹 󰁳󰁥

󰁤󰁥󰁮󰁯󰁴󰁡 󰁣󰁯󰁭󰁯 O(G).  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁔󰁡󰁭󰁡󰃱󰁯 󰁤󰁥 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯(󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯) G=(V,E) 󰁥󰁳 󰁥󰁬 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁥󰁪󰁥󰁳 (󰁡󰁲󰁣󰁯󰁳) 󰁱󰁵󰁥

󰁴󰁩󰁥󰁮󰁥, 󰁳󰁥 󰁤󰁥󰁮󰁯󰁴󰁡 󰁣󰁯󰁭󰁯 T(G)

 󰁅󰁪󰁥󰁭󰁰󰁬󰁯󰁳:

6

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

 󰁮  •  U󰁮 󰁧󰁲󰁡󰁦󰁯 󰁣󰁯󰁭󰁰󰁬󰁥󰁴󰁡󰁭󰁥󰁮󰁴󰁥 󰁳󰁩󰁭󰁰󰁬󰁥 󰁴󰁩󰁥󰁮󰁥     󰁤󰁥 󰁴󰁡󰁭󰁡󰃱󰁯, 󰁳󰁩󰁥󰁮󰁤󰁯 󰁮 󰁥󰁬 󰁯󰁲󰁤󰁥󰁮 󰁤󰁥󰁬  2  󰁧󰁲󰁡󰁦󰁯.  󰁮  •  U󰁮 󰁧󰁲󰁡󰁦󰁯 󰁣󰁯󰁭󰁰󰁬󰁥󰁴󰁯 󰁴󰁩󰁥󰁮󰁥   +󰁮 󰁤󰁯󰁮󰁤󰁥 󰁥󰁳󰁴󰁯󰁳 󰁮 󰁡 󰁭󰁡󰁹󰁯󰁲󰁥󰁳 󰁲󰁥󰁳󰁰󰁥󰁣󰁴󰁯 󰁡󰁬 󰁧󰁲󰁡󰁦󰁯 2  󰁣󰁯󰁭󰁰󰁬󰁥󰁴󰁡󰁭󰁥󰁮󰁴󰁥 󰁳󰁩󰁭󰁰󰁬󰁥 󰁳󰁯󰁮  󰁤󰁥󰁢󰁩󰁤󰁯󰁳 󰁡 󰁬󰁯󰁳 󰁢󰁵󰁣󰁬󰁥󰁳 󰁱󰁵󰁥 󰁨󰁡󰁹 󰁰󰁯󰁲 󰁣󰁡󰁤󰁡 󰁵󰁮󰁯 󰁤󰁥 󰁬󰁯󰁳 󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳.  󰁮  •  U󰁮 󰁧󰁲󰁡󰁦󰁯 󰁣󰁯󰁭󰁰󰁬󰁥󰁴󰁡󰁭󰁥󰁮󰁴󰁥 󰁳󰁩󰁭󰁰󰁬󰁥 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯 󰁴󰁩󰁥󰁮󰁥 2   =󰁮(󰁮󰀭1) 󰁤󰁥 󰁴󰁡󰁭󰁡󰃱󰁯, 󰁳󰁩󰁥󰁮󰁤󰁯 󰁮  2  󰁥󰁬 󰁯󰁲󰁤󰁥󰁮 󰁤󰁥󰁬 󰁧󰁲󰁡󰁦󰁯.  󰁮  •  U󰁮 󰁧󰁲󰁡󰁦󰁯 󰁣󰁯󰁭󰁰󰁬󰁥󰁴󰁯 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯 󰁴󰁩󰁥󰁮󰁥 2   +󰁮=󰁮2 󰁤󰁯󰁮󰁤󰁥 󰁥󰁳󰁴󰁯󰁳 󰁮 󰁡 󰁭󰁡󰁹󰁯󰁲󰁥󰁳 󰁲󰁥󰁳󰁰󰁥󰁣󰁴󰁯  2  󰁡󰁬 󰁡󰁮󰁴󰁥󰁲󰁩󰁯 󰁳󰁯󰁮 󰁤󰁥󰁢󰁩󰁤󰁯󰁳 󰁡 󰁬󰁯󰁳 󰁢󰁵󰁣󰁬󰁥󰁳 󰁱󰁵󰁥 󰁨󰁡󰁹 󰁰󰁯󰁲 󰁣󰁡󰁤󰁡 󰁵󰁮󰁯 󰁤󰁥 󰁬󰁯󰁳 󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳.

  󰁤󰁥 󰁇󰁲󰁡󰁤󰁯 󰁤󰁥 󰁩󰁮󰁣󰁩󰁤󰁥󰁮󰁣󰁩󰁡 󰁤󰁥 󰁵󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁩  󰁤󰁥

󰁵󰁮 󰁧󰁲󰁡󰁦󰁯(󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯) 󰁥󰁳 󰁥󰁬 󰁮󰂪 󰁤󰁥 󰁥󰁪󰁥󰁳(󰁡󰁲󰁣󰁯󰁳) 󰁥󰁮 󰁬󰁯󰁳 󰁱󰁵󰁥 󰁩󰁮󰁴󰁥󰁲󰁶󰁩󰁥󰁮󰁥 󰁥󰁬 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁩, 󰁣󰁯󰁮󰁳󰁩󰁤󰁥󰁲󰁡󰁮󰁤󰁯 󰁱󰁵󰁥 󰁥󰁬 󰁢󰁵󰁣󰁬󰁥 󰁻󰁩,󰁩󰁽 󰁣󰁵󰁥󰁮󰁴󰁡 󰁤󰁯󰁢󰁬󰁥. S󰁥 󰁤󰁥󰁮󰁯󰁴󰁡 󰁣󰁯󰁭󰁯 󰁧(󰁩). E󰁮 G󰁲󰁡󰁦󰁯󰁳 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯󰁳: •  󰁇󰁲󰁡󰁤󰁯 󰁤󰁥 󰁳󰁡󰁬󰁩󰁤󰁡 󰁧+(󰁩) 󰁥󰁳 󰁥󰁬 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁡󰁲󰁣󰁯󰁳 󰁤󰁯󰁮󰁤󰁥 󰁩 󰁥󰁸󰁴󰁲󰁥󰁭󰁯 󰁩󰁮󰁩󰁣󰁩󰁡󰁬. •  󰁇󰁲󰁡󰁤󰁯 󰁤󰁥 󰁥󰁮󰁴󰁲󰁡󰁤󰁡 󰁧󰀭(󰁩) 󰁥󰁳 󰁥󰁬 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁡󰁲󰁣󰁯󰁳 󰁤󰁯󰁮󰁤󰁥 󰁩 󰁥󰁳 󰁥󰁸󰁴󰁲󰁥󰁭󰁯 󰁦󰁩󰁮󰁡󰁬.  󰁐󰁲󰁯󰁰󰁩󰁥󰁤󰁡󰁤󰁥󰁳:

1.  E󰁬 󰁴󰁡󰁭󰁡󰃱󰁯 󰁤󰁥 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁥󰁳 󰁩󰁧󰁵󰁡󰁬 󰁡 󰁬󰁡 󰁳󰁥󰁭󰁩󰁳󰁵󰁭󰁡 󰁤󰁥 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁧󰁲󰁡󰁤󰁯󰁳 󰁤󰁥 󰁩󰁮󰁣󰁩󰁤󰁥󰁮󰁣󰁩󰁡: 1    󰁧 (󰁩) . E󰁳 󰁣󰁯󰁮󰁳󰁥󰁣󰁵󰁥󰁮󰁣󰁩󰁡 󰁤󰁥 󰁱󰁵󰁥 󰁣󰁡󰁤󰁡 󰁥󰁪󰁥 (󰁡󰁲󰁣󰁯) 󰁣󰁯󰁮󰁴󰁲󰁩󰁢󰁵󰁹󰁥 2 󰁶󰁥󰁣󰁥󰁳 󰁴 (󰁇 ) =  ∑ 2 󰁩∈󰁖  󰁥󰁮 󰁥󰁬 󰁧󰁲󰁡󰁤󰁯 󰁤󰁥 󰁩󰁮󰁣󰁩󰁤󰁥󰁮󰁣󰁩󰁡(󰁵󰁮󰁯 󰁰󰁯󰁲 󰁣󰁡󰁤󰁡 󰁶󰃩󰁲󰁴󰁩󰁣󰁥) 2.  E󰁮 󰁴󰁯󰁤󰁯 󰁧󰁲󰁡󰁦󰁯 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯 󰁹 󰁰󰁡󰁲󰁡 󰁴󰁯󰁤󰁯 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁩, 󰁧(󰁩)=󰁧+(󰁩)+󰁧󰀭(󰁩), 󰁹󰁡 󰁱󰁵󰁥 󰁴󰁡󰁮󰁴󰁯 󰁧+(󰁩) 󰁣󰁯󰁭󰁯 󰁧󰀭(󰁩) 󰁣󰁯󰁮󰁴󰁲󰁩󰁢󰁵󰁹󰁥󰁮 󰁥󰁮 󰁥󰁬 󰁧󰁲󰁡󰁤󰁯 󰁤󰁥 󰁩󰁮󰁣󰁩󰁤󰁥󰁮󰁣󰁩󰁡. 3.  E󰁮 󰁴󰁯󰁤󰁯 󰁧󰁲󰁡󰁦󰁯 󰁨󰁡󰁹 󰁵󰁮 󰁮󰂺󰁵󰁮󰁡 󰁰󰁡󰁲 󰁶󰁥󰁺 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁣󰁯󰁮 󰁧󰁲󰁡󰁤󰁯 󰁩󰁭󰁰󰁡󰁲:  󰁄󰁥󰁭󰁯󰁳󰁴󰁲󰁡󰁣󰁩󰃳󰁮: ∑ 󰁧 (󰁩) = 2󰁴 (󰁇)  󰁳󰁥󰁰󰁡󰁲󰁡󰁭󰁯󰁳 󰁧󰁲󰁡 󰁤󰁯󰁳  󰁰󰁡󰁲 󰁥 󰁩󰁭󰁰󰁡󰁲  󰁩∈󰁖 

󰂴󰁠

󰂴󰂴

󰂴󰂴

󰂴󰂴

󰁩∈󰁖 

󰁩∈󰁖 

󰁩∈󰁖 

󰁩∈󰁖 

∑ 󰁧 (󰁩) = ∑ 󰁧 (󰁩) + ∑ 󰁧 (󰁩) = 2󰁴 (󰁇) → ∑ 󰁧 (󰁩) 󰁳󰁵󰁭󰁡  󰁧 (󰁩) 󰁰󰁡󰁲  ∑ 󰁧 (󰁩) 󰁳󰁵󰁭󰁡  󰁧 (󰁩)󰁩󰁭󰁰󰁡󰁲  󰁩∈󰁖 

󰂴󰂴

󰂴

∑ 󰁧 (󰁩) = 2󰁴 (󰁇) − ∑ 󰁧 (󰁩) ≡  󰁰󰁡󰁲  󰁩∈󰁖 

󰁩∈󰁖 

123

 󰁰󰁡󰁲 

  󰁧(󰁩) 󰁥󰁳 󰁩󰁭󰁰󰁡󰁲, 󰁬󰁵󰁥󰁧󰁯 󰁰󰁡󰁲󰁡

󰂴󰂴 ∑ 󰁧(󰁩) 󰁩∈ V

󰁱󰁵󰁥 󰁳󰁥󰁡 󰁰󰁡󰁲 󰁤󰁥󰁢󰁥 󰁨󰁡󰁢󰁥󰁲 󰁵󰁮 󰁮󰁵󰁭󰁥󰁲󰁯 󰁰󰁡󰁲 󰁤󰁥 󰁧(󰁩).

7

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

1.7. CADENAS Y CICLOS.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: D󰁡󰁤󰁯 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 (E,V) 󰁬󰁬󰁡󰁭󰁡󰁭󰁯󰁳 󰁣󰁡󰁤󰁥󰁮󰁡  󰁡 󰁵󰁮󰁡 󰁳󰁥󰁣󰁵󰁥󰁮󰁣󰁩󰁡 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁬 󰁤󰁥

󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁹 󰁥󰁪󰁥󰁳 󰁱󰁵󰁥 󰁣󰁯󰁭󰁩󰁥󰁮󰁺󰁡 󰁹 󰁡󰁣󰁡󰁢󰁡 󰁥󰁮 󰁵󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥(󰁮󰁯 󰁮󰁥󰁣󰁥󰁳󰁡󰁲󰁩󰁡󰁭󰁥󰁮󰁴󰁥 󰁥󰁬 󰁭󰁩󰁳󰁭󰁯)  󰁅󰁪󰁥󰁭󰁰󰁬󰁯: 󰁶1,(󰁶1,󰁶2),󰁶2,(󰁶2,󰁶4),󰁶4, 󰁤󰁥󰁣󰁩󰁭󰁯󰁳 󰁱󰁵󰁥 󰁬󰁡 󰁣󰁡󰁤󰁥󰁮󰁡 󰁵󰁮󰁥 󰁶1 󰁣󰁯󰁮 󰁶4. T󰁩󰁰󰁯󰁳 󰁤󰁥 󰁣󰁡󰁤󰁥󰁮󰁡󰁳: •  C󰁡󰁤󰁥󰁮󰁡 󰁳󰁩󰁭󰁰󰁬󰁥 : 󰁮󰁯 󰁳󰁥 󰁲󰁥󰁰󰁩󰁴󰁥 󰁮󰁩󰁮󰁧󰃺󰁮 󰁥󰁪󰁥 •  C󰁡󰁤󰁥󰁮󰁡 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁬: N󰁯 󰁳󰁥 󰁲󰁥󰁰󰁩󰁴󰁥 󰁮󰁩󰁮󰁧󰃺󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁵󰁮 󰁣󰁩󰁣󰁬󰁯 󰁥󰁳 󰁵󰁮󰁡 󰁣󰁡󰁤󰁥󰁮󰁡 󰁤󰁯󰁮󰁤󰁥 󰁥󰁬 1󰂺 󰁹 󰁥󰁬 󰃺󰁬󰁴󰁩󰁭󰁯 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁥󰁳 󰁥󰁬 󰁭󰁩󰁳󰁭󰁯  󰁅󰁪󰁥󰁭󰁰󰁬󰁯: 󰁶1,(󰁶1,󰁶2),󰁶2,(󰁶2,󰁶4),󰁶4,(󰁶4,󰁶1),󰁶1. T󰁩󰁰󰁯󰁳: •  E󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁬 󰁳󰁩 󰁳󰁯󰁬󰁯 󰁣󰁯󰁩󰁮󰁣󰁩󰁤󰁥󰁮 󰁥󰁬 󰁰󰁲󰁩󰁭󰁥󰁲 󰁹 󰃺󰁬󰁴󰁩󰁭󰁯 󰁶󰃩󰁲󰁴󰁩󰁣󰁥.  󰁎󰁯󰁴󰁡: S󰁩 󰁴󰁲󰁡󰁢󰁡󰁪󰁡󰁭󰁯󰁳 󰁣󰁯󰁮 ,󰁭󰁵󰁬󰁴󰁩󰁧󰁲󰁡󰁦󰁯󰁳 󰁮󰁯 󰁤󰁥󰁢󰁥󰁭󰁯󰁳 󰁤󰁥󰁮󰁯󰁴󰁡󰁲 󰁬󰁯󰁳 󰁥󰁪󰁥󰁳 󰁣󰁯󰁭󰁯 (󰁶󰁩,󰁶󰁪), 󰁰󰁵󰁥󰁳

 󰁰󰁵󰁥󰁤󰁥 󰁯󰁣󰁵󰁲󰁲󰁩󰁲 󰁱󰁵󰁥 󰁣󰁯󰁮 󰁥󰁳󰁴󰁡 󰁮󰁯󰁴󰁡󰁣󰁩󰃳󰁮 󰁳󰁥 󰁥󰁮󰁣󰁵󰁥󰁮󰁴󰁲 󰁥󰁮󰁣󰁵󰁥󰁮󰁴󰁲󰁥󰁮 󰁥󰁮 󰁶󰁡󰁲󰁩󰁯󰁳 󰁥󰁪󰁥󰁳. T󰁥󰁮󰁤󰁲󰁥󰁭󰁯󰁳 󰁱󰁵󰁥 󰁲󰁥󰁰󰁲󰁥󰁳󰁥󰁮󰁴󰁡󰁲󰁬󰁯 󰁣󰁯󰁭󰁯 󰁥󰁬 󰁮󰁯󰁭󰁢󰁲󰁥 󰁤󰁥󰁬 󰁥󰁪󰁥. 1.8 CAMINOSY CIRCUITOS  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮 S󰁥󰁡 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯 (V,A), 󰁬󰁬󰁡󰁭󰁡󰁭󰁯󰁳 󰁣󰁡󰁭󰁩󰁮󰁯 󰁡 󰁵󰁮󰁡 󰁳󰁥󰁣󰁵󰁥󰁮󰁣󰁩󰁡 󰁤󰁥 󰁡󰁲󰁣󰁯󰁳 󰁡1,󰁡2...󰁤󰁯󰁮󰁤󰁥 󰁥󰁬 :󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁦󰁩󰁮󰁡󰁬 󰁤󰁥 󰁣󰁡󰁤󰁡 󰁡󰁲󰁣󰁯 󰁣󰁯󰁩󰁮󰁣󰁩󰁤󰁥 󰁣󰁯󰁮 󰁥󰁬 󰁩󰁮󰁩󰁣󰁩󰁡󰁬 󰁤󰁥󰁬 󰁳󰁩󰁧󰁵󰁩󰁥󰁮󰁴󰁥: 󰁦󰁩󰁮󰁡󰁬(󰁡󰁩)=󰁯󰁲󰁩󰁧󰁥󰁮(󰁡󰁩+1) ∀󰁩.

T󰁩󰁰󰁯󰁳 : •  C󰁡󰁭󰁩󰁮󰁯 󰁳󰁩󰁭󰁰󰁬󰁥. N󰁯 󰁳󰁥 󰁲󰁥󰁰󰁩󰁴󰁥 󰁮󰁩󰁮󰁧󰃺󰁮 󰁡󰁲󰁣󰁯. •  C󰁡󰁭󰁩󰁮󰁯 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁬: N󰁯 󰁳󰁥 󰁲󰁥󰁰󰁩󰁴󰁥 󰁮󰁩󰁮󰁧󰃺󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁃󰁩󰁲󰁣󰁵󰁩󰁴󰁯 󰁥󰁳 󰁵󰁮 󰁣󰁡󰁭󰁩󰁮󰁯 󰁤󰁯󰁮󰁤󰁥 󰁥󰁬 󰁰󰁲󰁩󰁭󰁥󰁲 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁣󰁯󰁩󰁮󰁣󰁩󰁤󰁥 󰁣󰁯󰁮 󰁥󰁬 󰁦󰁩󰁮󰁡󰁬

󰁥3 A

󰁥1

󰁥2

B

C

󰁥4 󰁥5 S󰁩󰁭󰁰󰁬󰁥: 󰁥1,󰁥2,󰁥4 E󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁬: 󰁥1,󰁥2, C󰁩󰁲󰁣󰁵󰁩󰁴󰁯: 󰁥1,󰁥2,󰁥5; 󰁥2,󰁥4; 󰁥3,󰁥4.

8

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

1.9. SUBGRAFOS  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁤󰁩󰁲󰁥󰁭󰁯󰁳 󰁱󰁵󰁥 (V󰂴,E󰂴) 󰁥󰁳 󰁵󰁮 󰁳󰁵󰁢󰁧󰁲󰁡󰁦󰁯 󰁤󰁥 (V,E) 󰁳󰁩 󰁳󰁥 󰁣󰁵󰁭󰁰󰁬󰁥 󰁱󰁵󰁥 V󰂴⊆V 󰁹

E󰂴⊆E.  󰁅󰁪󰁥󰁭󰁰󰁬󰁯: G1=(V1, E1), V1=(1,2,3); E1=󰁻(1,2),(2,3)󰁽

G2=(V2,E2), E2=󰁻(1,2)󰁽 G2 󰁳󰁵󰁢󰁧󰁲󰁡󰁦󰁯 󰁤󰁥V2=(1,2); G1  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: D󰁩󰁲󰁥󰁭󰁯󰁳 󰁱󰁵󰁥 (V󰂴,E󰂴) 󰁥󰁳 󰁥󰁬  󰁧󰁲󰁡󰁦󰁯 󰁣󰁯󰁭󰁰󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁲󰁩󰁯 󰁣󰁯󰁭󰁰󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁲󰁩󰁯 󰁤󰁥 (V,E) 󰁳󰁩 󰁳󰁥 󰁣󰁵󰁭󰁰󰁬󰁥

󰁱󰁵󰁥 V=V󰂴󰁹 E󰂴=E󰁣(󰁣󰁯󰁭󰁰󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁲󰁩󰁯 󰁤󰁥 E).  󰁅󰁪󰁥󰁭󰁰󰁬󰁯: G=󰁻V=(1,2,3);E=󰁻(1,2),(2,3),(1,1)󰁽󰁽 E󰂴=󰁻(1,3󰁽,(2,2),(3,3)󰁽   G󰂴=󰁻V󰂴=(1,2,3); E󰂴=󰁻(1,3󰁽,(2,2),(3,3)󰁽

1

2

3

 󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮 G󰁲󰁡󰁦󰁯 󰁤󰁥 󰁵󰁮󰁩󰃳󰁮  󰁤󰁥 󰁤󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 (V,E), (V󰂴,E󰂴) 󰁥󰁳 󰁥󰁬 󰁧󰁲󰁡󰁦󰁯 󰁲󰁥󰁳󰁵󰁬󰁴󰁡󰁮󰁴󰁥 󰁤󰁥 󰁵󰁮󰁩󰁲 V 󰁹 V󰂴(V∪:V󰂴),󰁹 E 󰁹 E󰂴 (E∪E󰂴)  󰁅󰁪󰁥󰁭󰁰󰁬󰁯: G1=󰁻V1=(1,2,3); E1=󰁻(1,2)󰁽󰁽 G2=(V2=(2,3,4); E2=󰁻(2,4)󰁽 G=G1∪G2=󰁻V=(1,2,3,4),E=󰁻(1,2󰁽,(2,4)󰁽

2.󰀭󰁇󰁒󰁁󰁆󰁏󰁓 󰁃󰁏󰁎󰁅󰁘󰁏󰁓 2.1 DEFINICION GRAFOS CONEXOS  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: G󰁲󰁡󰁦󰁯 G=(V,E) 󰁥󰁳 󰁣󰁯󰁮󰁥󰁸󰁯 󰁳󰁩 󰁮󰁯 󰁳󰁥 󰁰󰁵󰁥󰁤󰁥 󰁰󰁯󰁮󰁥󰁲 󰁣󰁯󰁭󰁯 󰁵󰁮󰁩󰃳󰁮 󰁤󰁥 2 󰁧󰁲󰁡󰁦󰁯󰁳

󰁤󰁩󰁳󰁪󰁵󰁮󰁴󰁯󰁳 (󰁮󰁯 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁥󰁮 󰁣󰁯󰁭󰃺󰁮), 󰁥󰁳󰁴󰁯 󰁯󰁣󰁵󰁲󰁲󰁥 󰁳󰁩 󰁥󰁮 󰁥󰁬 󰁧󰁲󰁡󰁦󰁯 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁵󰁮󰁩󰁤󰁯󰁳 󰁦󰁯󰁲󰁭󰁡󰁮󰁤󰁯 󰁵󰁮 󰁢󰁬󰁯󰁱󰁵󰁥.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁧󰁲󰁡󰁦󰁯 󰁩󰁮󰁣󰁯󰁮󰁥󰁸󰁯 󰁥󰁳 󰁡󰁱󰁵󰁥󰁬 󰁱󰁵󰁥 󰁮󰁯 󰁥󰁳 󰁣󰁯󰁮󰁥󰁸󰁯.  󰁅󰁪󰁥󰁭󰁰󰁬󰁯:

1

2

3 C󰁯󰁮󰁥󰁸󰁯

4 I󰁮󰁣󰁯󰁮󰁥󰁸󰁯(󰁧󰁥󰁮󰁥󰁲󰁡󰁤󰁯 󰁤󰁯󰁳 󰁣󰁯󰁭󰁰󰁯󰁮󰁥󰁮󰁴󰁥󰁳 󰁣󰁯󰁮󰁥󰁸󰁡󰁳) 1

2

3

󰁰󰁯󰁲

4

9

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

  2.2.CARACTERIZACI󰃓N DE LOS GRAFOS CONEXOS.GRAFO ORIENTADO FUERTEMENTE CONEXO 󰁥󰁳 󰁣󰁯󰁮󰁥󰁸󰁯 P󰁯󰁤󰁥󰁭󰁯󰁳 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁣󰁯󰁮󰁥󰁸󰁯󰁳󰁥󰁸󰁩󰁳󰁴󰁥 󰁭󰁥󰁤󰁩󰁡󰁮󰁴󰁥 󰁣󰁡󰁤󰁥󰁮󰁡󰁳: 󰁳󰁩 󰁹 󰁳󰃳󰁬󰁯 󰁳󰁩 󰁰󰁡󰁲󰁡󰁣󰁡󰁲󰁡󰁣󰁴󰁥󰁲󰁩󰁺󰁡󰁲 󰁣󰁵󰁡󰁬󰁱󰁵󰁩󰁥󰁲 󰁰󰁡󰁲󰁥󰁪󰁡 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁵󰁮󰁡 󰁬󰁡󰁳 󰁣󰁡󰁤󰁥󰁮󰁡 󰁱󰁵󰁥 U󰁮 󰁬󰁯󰁳  󰁧󰁲󰁡󰁦󰁯 󰁵󰁮󰁥 󰁳󰁩󰁥󰁮󰁤󰁯 󰁥󰁳󰁴󰁡   󰁳󰁩󰁭󰁰󰁬󰁥 󰁹 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁬.

 󰁄󰁥󰁭󰁯󰁳󰁴󰁲󰁡󰁣󰁩󰃳󰁮:

1)S󰁩 󰁵󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁮󰁯 󰁵󰁮󰁩󰁤󰁯 󰁣󰁯󰁮 󰁯󰁴󰁲󰁯 󰁨󰁡󰁢󰁲󰃡 󰁵󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥( 󰁮󰁯 󰁦󰁩󰁮󰁡󰁬 󰁮󰁩 󰁩󰁮󰁩󰁣󰁩󰁡󰁬 󰁤󰁥 󰁬󰁡 󰁣󰁡󰁤󰁥󰁮󰁡) 󰁱󰁵󰁥 󰁳󰃳󰁬󰁯 󰁵󰁮󰁩󰁤󰁯 󰁣󰁯󰁮 󰁯󰁴󰁲󰁯 󰁶󰃩󰁲󰁴󰁩󰁣󰁥, 󰁹 󰁮󰁯 󰁳󰁥󰁲󰃡 󰁣󰁯󰁮󰁥󰁸󰁯. 2)E󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁬: 󰁳󰁩 󰁥󰁮 󰁬󰁡 󰁣󰁡󰁤󰁥󰁮󰁡 󰁳󰁥 󰁲󰁥󰁰󰁩󰁴󰁥 󰁵󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁥󰁬󰁩󰁭󰁩󰁮󰁡󰁭󰁯󰁳 󰁥󰁬 󰁣󰁩󰁣󰁬󰁯 󰁱󰁵󰁥 󰁥󰁭󰁰󰁩󰁥󰁺󰁡 󰁹 󰁡󰁣󰁡󰁢 󰁥󰁮 󰁥󰁳󰁴󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥., 󰁹 󰁡󰁨󰁯󰁲󰁡 󰁬󰁡 󰁣󰁡󰁤󰁥󰁮󰁡 󰁳󰁥󰁲󰃡 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁡󰁬. 3) S󰁩󰁭󰁰󰁬󰁥: 󰁤󰁥 󰁩󰁧󰁵󰁡󰁬 󰁦󰁯󰁲󰁭󰁡 󰁳󰁩 󰁮󰁯 󰁥󰁳 󰁳󰁩󰁭󰁰󰁬󰁥 󰁥󰁳 󰁰󰁯󰁲󰁱󰁵󰁥 󰁨󰁡󰁹 󰁵󰁮 󰁢󰁵󰁣󰁬󰁥 󰁱󰁵󰁥 󰁰󰁯󰁤󰁥󰁭󰁯󰁳 󰁥󰁬󰁩󰁭󰁩󰁮󰁡󰁲 󰁹 󰁬󰁡 󰁣󰁡󰁤󰁥󰁮󰁡 󰁱󰁵󰁥 󰁮󰁯󰁳 󰁱󰁵󰁥󰁤󰁡 󰁥󰁳 󰁳󰁩󰁭󰁰󰁬󰁥.  󰁅󰁪󰁥󰁭󰁰󰁬󰁯:

1

2 󰁥1

󰁥3󰁥2 4  5  6 

3

1󰀭2󰁥1 1󰀭3 󰁥1,󰁥2 1󰀭4 󰁥1,󰁥4 2󰀭3󰁥2 2󰀭4󰁥3 3󰀭4󰁥2,󰁥3

E󰁮 󰁧󰁲󰁡󰁦󰁯󰁳 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯󰁳 󰁬󰁡 󰁮󰁵󰁥󰁶󰁡 󰁤󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮 󰁮󰁯 󰁶󰃡󰁬󰁩󰁤󰁡 󰁰󰁵󰁥󰁳 󰁰󰁵󰁥󰁤󰁥󰁮 󰁳󰁥󰁲 󰁣󰁯󰁮󰁥󰁸󰁯󰁳 󰁹 󰁮󰁯 󰁨󰁡󰁢󰁥󰁲 󰁣󰁡󰁭󰁩󰁮󰁯 󰁥󰁮󰁴󰁲󰁥 󰁤󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 (󰁤󰁥󰁢󰁩󰁤󰁯 󰁡󰁬 󰁳󰁥󰁮󰁴󰁩󰁤󰁯)  󰁅󰁪󰁥󰁭󰁰󰁬󰁯: 󰁥󰁳 󰁣󰁯󰁮󰁥󰁸󰁯 󰁹 3 󰁮󰁯 󰁳󰁥 󰁰󰁵󰁥󰁤󰁥 󰁵󰁮󰁩󰁲 󰁣󰁯󰁮 1, 󰁮󰁩 󰁣󰁯󰁮 2. 1 2 3  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁤󰁩󰁲󰁩󰁧󰁩󰁤󰁯 󰁥󰁳   󰁦󰁵󰁥󰁲󰁴󰁥󰁭󰁥󰁮󰁴󰁥 󰁣󰁯󰁮󰁥󰁸󰁯 󰁳󰁩 󰁨󰁡󰁹 󰁡󰁬󰁧󰃺󰁮 󰁣󰁡󰁭󰁩󰁮󰁯 󰁱󰁵󰁥 󰁵󰁮󰁥

󰁴󰁯󰁤󰁯󰁳 󰁳󰁵󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳. 󰁣󰁯󰁮󰁣󰁥󰁰󰁴󰁯 󰁥󰁳󰁴󰃡󰁮 󰁵󰁮󰁩󰁤󰁯󰁳, 󰁹󰁡 󰁱󰁵󰁥E󰁳󰁴󰁥 󰁨󰁡󰁹 󰁣󰁡󰁬󰁬󰁥󰁳 󰁣󰁯󰁮 󰁩󰁮󰁴󰁥󰁲󰃩󰁳 󰁳󰁥󰁮󰁴󰁩󰁤󰁯 󰁰󰁡󰁲󰁡 󰃺󰁮󰁩󰁣󰁯.󰁳󰁡󰁢󰁥󰁲 󰁳󰁩 󰁥󰁮 󰁵󰁮󰁡 󰁣󰁩󰁵󰁤󰁡󰁤 󰁴󰁯󰁤󰁯󰁳 󰁳󰁵󰁳 󰁲󰁩󰁮󰁣󰁯󰁮󰁥󰁳  󰁅󰁪󰁥󰁭󰁰󰁬󰁯:

10

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

 

3.  󰁇󰁒󰁁󰁆󰁏 󰁅󰁕󰁌󰁅󰁒󰁉󰁁󰁎󰁏, 󰁐󰁕󰁅󰁎󰁔󰁅󰁓 󰁄󰁅 󰁋󰃖󰁎󰁉󰁓󰁇󰁕󰁅󰁒. E󰁮 󰁥󰁬 󰁳󰁩󰁧󰁬󰁯 XVIII E󰁵󰁬󰁥󰁲 󰁤󰁥󰁭󰁯󰁳󰁴󰁲󰃳 󰁬󰁡 󰁩󰁭󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤 󰁤󰁥 󰁲󰁥󰁣󰁯󰁲󰁲󰁥󰁲 󰁬󰁯󰁳 7 󰁰󰁵󰁥󰁮󰁴󰁥󰁳 󰁤󰁥 󰁳󰁵 󰁣󰁩󰁵󰁤󰁡󰁤 󰁳󰁩󰁮 󰁰󰁡󰁳󰁡󰁲 󰁤󰁯󰁳 󰁶󰁥󰁣󰁥󰁳 󰁰󰁯󰁲 󰁥󰁬 󰁭󰁩󰁳󰁭󰁯 󰁰󰁵󰁥󰁮󰁴󰁥. A 󰁰󰁡󰁲󰁴󰁩󰁲 󰁤󰁥 󰁡󰁱󰁵󰃭 󰁳󰁥 󰁣󰁲󰁥󰁡 󰁵󰁮 󰁴󰁩󰁰󰁯 󰁤󰁥 󰁧󰁲󰁡󰁦󰁯󰁳, 󰁬󰁯󰁳 E󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯󰁳. 3.1 EL PROBLEMA DE LOS PUENTES DE K󰃖NISGUER E󰁵󰁬󰁥󰁲 󰁤󰁥󰁭󰁯󰁳󰁴󰁲󰃳 󰁥󰁬 󰁳󰁩󰁧󰁵󰁩󰁥󰁮󰁴󰁥 󰁴󰁥󰁯󰁲󰁥󰁭󰁡 󰁔󰁥󰁯󰁲󰁥󰁭󰁡󰀺 󰁴󰁯󰁤󰁯 󰁧󰁲󰁡󰁦󰁯 󰁥󰁳 󰁲󰁥󰁣󰁯󰁲󰁲󰁩󰁢󰁬󰁥 󰁳󰁩󰁮 󰁲󰁥󰁰󰁥󰁴󰁩󰁣󰁩󰃳󰁮 󰁤󰁥 󰁮󰁩󰁮󰁧󰁵󰁮󰁯 󰁤󰁥 󰁳󰁵󰁳 󰁡󰁲󰁣󰁯󰁳, 󰁳󰁩 󰁥󰁬 󰁮󰁵󰁭󰁥󰁲󰁯 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁰󰁡󰁲󰁥󰁳( 󰁬󰁬󰁥󰁧󰁡󰁮 󰁵󰁮 󰁮󰂺 󰁩󰁭󰁰󰁡󰁲 󰁤󰁥 󰁥󰁪󰁥󰁳) 󰁮󰁯 󰁥󰁳 󰁭󰁡󰁹󰁯󰁲 󰁱󰁵󰁥 󰀲, 󰁥󰁳 󰁤󰁥󰁣󰁩󰁲 󰁱󰁵󰁥 󰁮󰁯 󰁳󰁥󰁡󰁮 4, 6,8...(󰁹󰁡 󰁱󰁵󰁥 󰁣󰁯󰁭󰁯 󰁶󰁩󰁭󰁯󰁳 󰁥󰁮 󰁵󰁮 󰁴󰁥󰁯󰁲󰁥󰁭󰁡 󰁡󰁮󰁴󰁥󰁲󰁩󰁯󰁲, 󰁨󰁡󰁹 󰁵󰁮 󰁮󰁵󰁭󰁥󰁲󰁯 󰁰󰁡󰁲 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁰󰁡󰁲󰁥󰁳)  󰁄󰁥󰁭󰁯󰁳󰁴󰁲󰁡󰁣󰁩󰃳󰁮  S󰁩 󰁵󰁮 󰁣󰁡󰁭󰁩󰁮󰁯 󰁮󰁯 󰁥󰁭󰁰󰁩󰁥󰁺󰁡 󰁹 󰁴󰁥󰁲󰁭󰁩󰁮󰁡 󰁥󰁮 󰁵󰁮 󰁭󰁩󰁳󰁭󰁯 󰁶󰃩󰁲󰁴󰁩󰁣󰁥, 󰁣󰁡󰁤󰁡 󰁶󰁥󰁺

󰁱󰁵󰁥 󰁬󰁬󰁥󰁧󰁡 󰁡 󰁵󰁮 󰁥󰁪󰁥, 󰁥󰁳󰁴󰁥 󰁨󰁡 󰁤󰁥 󰁳󰁡󰁬󰁩󰁲 󰁰󰁯󰁲 󰁯󰁴󰁲󰁯 󰁥󰁪󰁥 󰁮󰁯 󰁵󰁴󰁩󰁬󰁩󰁺󰁡󰁤󰁯. D󰁥 󰁥󰁳󰁴󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁨󰁡󰁮 󰁤󰁥 󰁳󰁡󰁬󰁩󰁲 󰁥󰁮󰁴󰁯󰁮󰁣󰁥󰁳 󰁵󰁮 󰁮󰁵󰁭󰁥󰁲󰁯 󰁰󰁡󰁲 󰁤󰁥 󰁡󰁲󰁣󰁯󰁳, 󰁬󰁵󰁥󰁧󰁯 󰁥󰁳 󰁵󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁰󰁡󰁲. P󰁵󰁥󰁤󰁥 󰁨󰁡󰁢󰁥󰁲 󰁤󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁰󰁡󰁲󰁥󰁳, 󰁥󰁮 󰁤󰁯󰁮󰁤󰁥 󰁥󰁬 󰁣󰁡󰁭󰁩󰁮󰁯 󰁥󰁭󰁰󰁩󰁥󰁺󰁡 󰁹 󰁴󰁥󰁲󰁭󰁩󰁮󰁡, 󰁰󰁥󰁲󰁯 󰁮󰁯 󰁰󰁵󰁥󰁤󰁥 󰁨󰁡󰁢󰁥󰁲 󰁭󰁡󰁳 󰁤󰁥 󰁤󰁯󰁳 (󰁳󰃳󰁬󰁯 󰁨󰁡󰁹 󰁵󰁮 󰁣󰁯󰁭󰁩󰁥󰁮󰁺󰁯 󰁹 󰁵󰁮 󰁦󰁩󰁮󰁡󰁬), 󰁰󰁵󰁥󰁳 󰁳󰁩󰁮󰁯 󰁴󰁥󰁮󰁤󰁲󰃭󰁡󰁮 󰁱󰁵󰁥 󰁰󰁡󰁳󰁡󰁲󰁳󰁥 󰁰󰁯󰁲 󰁡󰁬󰁧󰃺󰁮 󰁥󰁪󰁥 󰁭󰃡󰁳 󰁤󰁥 󰁵󰁮󰁡 󰁶󰁥󰁺 󰁰󰁡󰁲󰁡  󰁰󰁡󰁳󰁡󰁲 󰁰󰁯󰁲 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳. S󰁩 󰁮󰁯 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁰󰁡󰁲󰁥󰁳, 󰁥󰁮󰁴󰁯󰁮󰁣󰁥󰁳 󰁥󰁬 󰁲󰁥󰁣󰁯󰁲󰁲󰁩󰁤󰁯 󰁮󰁯 󰁴󰁩󰁥󰁮󰁥 󰁰󰁲󰁩󰁮󰁣󰁩󰁰󰁩󰁯 󰁮󰁩 󰁦󰁩󰁮,  󰁰󰁵󰁤󰁩󰁥󰁮󰁤󰁯 󰁥󰁭󰁰󰁥󰁺󰁡󰁲 󰁹 󰁡󰁣󰁡󰁢󰁡󰁲 󰁥󰁮 󰁣󰁵󰁡󰁬󰁱󰁵󰁩󰁥󰁲 󰁣󰁵 󰁡󰁬󰁱󰁵󰁩󰁥󰁲 󰁶󰃩󰁲󰁴󰁩󰁣󰁥.  󰁅󰁪󰁥󰁭󰁰󰁬󰁯󰁳:

󰁁

B 1

(󰁧󰁲󰁡󰁦󰁯1)

(󰁧󰁲󰁡󰁦󰁯2)  C

5 2

6 E

C

3

4 A

󰁄 L󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 A 󰁹 D 󰁳󰁯󰁮 󰁩󰁭󰁰󰁡󰁲󰁥󰁳 󰁹 󰁥󰁮 󰁥󰁬󰁬󰁯󰁳 󰁥󰁭󰁰󰁩󰁥󰁺󰁡 󰁹 󰁴󰁥󰁲󰁭󰁩󰁮󰁡 󰁥󰁬 󰁣󰁩󰁲󰁣󰁵󰁩󰁴󰁯.

B 

C󰁯󰁭󰁯 󰁮󰁯 󰁴󰁩󰁥󰁮󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁰󰁡󰁲󰁥󰁳 󰁮󰁯 󰁴󰁩󰁥󰁮󰁥 󰁮󰁩 󰁰󰁲󰁩󰁮󰁣󰁩󰁰󰁩󰁯 󰁮󰁩 󰁦󰁩󰁮

11

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

(󰁇󰁲󰁡󰁦󰁯 3) 1  󰁄 5

2,6

󰁁 8 

󰁂 4

3,7

󰁃 P󰁵󰁥󰁮󰁴󰁥 󰁤󰁥 K󰃶󰁮󰁩󰁳󰁧󰁵󰁥󰁲 󰁤󰁯󰁮󰁤󰁥 󰁨󰁡󰁹 4 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁰󰁡󰁲󰁥󰁳, 󰁬󰁵󰁥󰁧󰁯 󰁮󰁯 󰁰󰁵󰁥󰁤󰁥 󰁳󰁥󰁲 󰁲󰁥󰁣󰁯󰁲󰁲󰁩󰁤󰁯 󰁳󰁩󰁮 󰁲󰁥󰁰󰁥󰁴󰁩󰁣󰁩󰃳󰁮 󰁤󰁥 󰁡󰁬󰁧󰁵󰁮󰁯 󰁤󰁥 󰁳󰁵󰁳 󰁥󰁪󰁥󰁳, 󰁣󰁯󰁭󰁯 󰁥󰁳 󰁬󰁥 󰁣󰁡󰁳󰁯 󰁤󰁥 󰁬󰁯󰁳 󰁥󰁪󰁥󰁳 󰁻B,C󰁽 󰁹 󰁻B,D󰁽

3.2. GRAFO EULERIANO Y SEMIEULERANO  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁃󰁯󰁬󰁡 󰁥󰁳 󰁵󰁮󰁡 󰁳󰁥󰁣󰁵󰁥󰁮󰁣󰁩󰁡 󰁤󰁥 󰁡󰁲󰁩󰁳󰁴󰁡󰁳 V0, V1,....,V󰁮󰀭1, V󰁮, 󰁥󰁮 󰁬󰁡 󰁱󰁵󰁥 󰁴󰁯󰁤󰁡󰁳 󰁬󰁡󰁳 󰁡󰁲󰁩󰁳󰁴󰁡󰁳 󰁳󰁯󰁮 󰁤󰁩󰁦󰁥󰁲󰁥󰁮󰁴󰁥󰁳. S󰁩 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁦󰁩󰁮󰁡󰁬 󰁥 󰁩󰁮󰁩󰁣󰁩󰁡󰁬 󰁣󰁯󰁩󰁮󰁣󰁩󰁤󰁥󰁮, 󰁳󰁥 󰁤󰁩󰁣󰁥 󰁱󰁵󰁥 󰁥󰁳 󰁵󰁮󰁡 󰁣󰁯󰁬󰁡  󰁣󰁥󰁲󰁲󰁡󰁤󰁡.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁳󰁥 󰁤󰁩󰁣󰁥 󰁧󰁲󰁡󰁦󰁯

󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯 󰁡 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁣󰁯󰁮󰁥󰁸󰁯 G 󰁱󰁵󰁥 󰁴󰁩󰁥󰁮󰁥 󰁵󰁮󰁡 󰁣󰁯󰁬󰁡 󰁣󰁥󰁲󰁲󰁡󰁤󰁡 󰁱󰁵󰁥 󰁩󰁮󰁣󰁬󰁵󰁹󰁥 󰁴󰁯󰁤󰁡󰁳 󰁬󰁡󰁳 󰁡󰁲󰁩󰁳󰁴󰁡󰁳 󰁤󰁥 G  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁳󰁥 󰁤󰁩󰁣󰁥 󰁧󰁲󰁡󰁦󰁯 󰁳󰁥󰁭󰁩󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯 󰁳󰁩 󰁬󰁡 󰁣󰁯󰁬󰁡 󰁮󰁯 󰁥󰁳 󰁣󰁥󰁲󰁲󰁡󰁤󰁡.

󰁔󰁥󰁯󰁲󰁥󰁭󰁡 󰁤󰁥 󰁅󰁵󰁬󰁥󰁲󰀺 1.  U󰁮 󰁧󰁲󰁡󰁦󰁯 󰁥󰁳 󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯 󰁳󰁩 󰁹 󰁳󰁯󰁬󰁯 󰁳󰁩 󰁣󰁡󰁤󰁡 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁥󰁳 󰁤󰁥 󰁧󰁲󰁡󰁤󰁯 󰁰󰁡󰁲. 2.  U󰁮 󰁧󰁲󰁡󰁦󰁯 󰁥󰁳 󰁳󰁥󰁭󰁩󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯 󰁳󰁩 󰁹 󰁳󰁯󰁬󰁯 󰁳󰁩 󰁴󰁩󰁥󰁮󰁥 󰁤󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁰󰁡󰁲󰁥󰁳 D󰁥󰁭󰁯󰁳󰁴󰁲󰁡󰁣󰁩󰃳󰁮: 1.    S󰁥󰁡 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯, 󰁥󰁮󰁴󰁯󰁮󰁣󰁥󰁳 󰁴󰁩󰁥󰁮󰁥 󰁵󰁮󰁡 󰁣󰁯󰁬󰁡 󰁣󰁥󰁲󰁲󰁡󰁤󰁡, 󰁬󰁵󰁥󰁧󰁯 󰁤󰁥 󰁣󰁡󰁤󰁡 󰁡󰁲󰁩󰁳󰁴󰁡 󰁥󰁮󰁴󰁲󰁡 󰁥󰁬 󰁭󰁩󰁳󰁭󰁯 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁡󰁲󰁣󰁯󰁳 󰁱󰁵󰁥 󰁳󰁡󰁬󰁥󰁮, 󰁹 󰁣󰁯󰁭󰁯 󰁥󰁳󰁴󰁯󰁳 󰁡󰁲󰁣󰁯󰁳 󰁮󰁯 󰁥󰁳󰁴󰃡󰁮 󰁲󰁥󰁰󰁥󰁴󰁩󰁤󰁯󰁳, 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁳󰁯󰁮 󰁰󰁡󰁲󰁥󰁳.   S󰁥󰁡

󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁳󰁩󰁮 󰁶󰁥󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁰󰁡󰁲󰁥󰁳, 󰁣󰁯󰁭󰁯 󰁶󰁩󰁭󰁯󰁳 󰁥󰁳󰁴󰁥 󰁳󰁥 󰁰󰁵󰁥󰁤󰁥 󰁲󰁥󰁣󰁯󰁲󰁲󰁥󰁲  󰁰󰁡󰁳󰁡󰁮󰁤󰁯 󰁰󰁯󰁲 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁡󰁲󰁣󰁯󰁳 󰁳󰁩󰁮 󰁲󰁥󰁰󰁥󰁴󰁩󰁣󰁩󰃳󰁮 󰁤󰁥 󰁮󰁩󰁮󰁧󰁵󰁮󰁯, 󰁤󰁥 󰁦󰁯󰁲󰁭󰁡 󰁱󰁵󰁥 󰁥󰁬 󰁣󰁩 󰁣󰁩󰁲󰁣󰁵󰁩󰁴󰁯 󰁲󰁣󰁵󰁩󰁴󰁯 󰁮󰁯 󰁴󰁩󰁥󰁮󰁥 󰁰󰁲󰁩󰁮󰁣󰁩󰁰󰁩󰁯 󰁮󰁩 󰁦󰁩󰁮, 󰁥󰁳 󰁤󰁥󰁣󰁩󰁲 󰁥󰁳 󰁵󰁮 󰁲󰁥󰁣󰁯󰁲󰁲󰁩󰁤󰁯 󰁣󰁥󰁲󰁲󰁡󰁤󰁯., 󰁬󰁵󰁥󰁧󰁯 󰁴󰁩󰁥󰁮󰁥 󰁵󰁮󰁡 󰁣󰁯󰁬󰁡 󰁣󰁥󰁲󰁲󰁡󰁤󰁡 󰁹 󰁰󰁯󰁲 󰁴󰁡󰁮󰁴󰁯 󰁥󰁳 󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯.

12

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

2.  S󰁥󰁡 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁳󰁥󰁭󰁩󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯, 󰁥󰁮󰁴󰁯󰁮󰁣󰁥󰁳 󰁴󰁩󰁥󰁮󰁥 󰁵󰁮󰁡 󰁣󰁯󰁬󰁡 󰁮󰁯 󰁣󰁥󰁲󰁲󰁡󰁤󰁡, 󰁬󰁵󰁥󰁧󰁯 󰁨󰁡󰁹 󰁵󰁮 󰁶󰁥󰁲󰁴󰁩󰁣󰁥 󰁤󰁯󰁮󰁤󰁥 󰁥󰁭󰁰󰁩󰁥󰁺󰁡 󰁬󰁡 󰁣󰁯󰁬󰁡, 󰁱󰁵󰁥 󰁴󰁩󰁥󰁮󰁥 󰁵󰁮 󰁮󰂺 󰁩󰁭󰁡󰁰󰁡󰁲 󰁤󰁥 󰁡󰁲󰁣󰁯󰁳, 󰁰󰁵󰁥󰁳 󰁤󰁥 󰃩󰁬 󰁳󰁡󰁬󰁥 󰁵󰁮󰁯 󰁭󰃡󰁳 󰁱󰁵󰁥 󰁥󰁮󰁴󰁲󰁡; 󰁤󰁥 󰁤󰁥 󰁩󰁧󰁵󰁡󰁬 󰁦󰁯󰁲󰁭󰁡 󰁴󰁥󰁮󰁤󰁲󰁡 󰁵󰁮 󰁶󰁥󰁲󰁴󰁩󰁣󰁥 󰁦󰁩󰁮󰁡󰁬 󰁤󰁯󰁮󰁤󰁥 󰁬󰁬󰁥󰁧󰁵󰁥 󰁵󰁮 󰁵󰁮 󰁡󰁲󰁣󰁯 󰁭󰁡󰁳 󰁱󰁵󰁥 󰁳󰁡󰁬󰁧󰁡, 󰁬󰁵󰁥󰁧󰁯 󰁴󰁡󰁭󰁢󰁩󰁥󰁮 󰁳󰁥󰁲󰁡 󰁩󰁭󰁰󰁡󰁲. E󰁬 󰁲󰁥󰁳󰁴󰁯 󰁤󰁥 󰁶󰁥󰁲󰁴󰁩󰁣󰁥󰁳 󰁳󰁡󰁬󰁥󰁮 󰁥󰁬 󰁭󰁩󰁳󰁭󰁯 󰁮󰁵󰁭󰁥󰁲󰁯 󰁤󰁥 󰁡󰁲󰁣󰁯󰁳 󰁱󰁵󰁥 󰁬󰁬󰁥󰁧󰁡󰁮, 󰁰󰁯󰁲 󰁬󰁯 󰁱󰁵󰁥 󰁳󰁯󰁮 󰁰󰁡󰁲󰁥󰁳. A󰁳󰁩 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 󰁱󰁵󰁥 󰁴󰁯󰁤󰁯 󰁧󰁲󰁡󰁦󰁯 󰁳󰁥󰁭󰁩󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯 󰁴󰁩󰁥󰁮󰁥 2 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁡󰁰󰁡󰁲󰁥󰁳.   S󰁩 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁣󰁯󰁮 2 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁰󰁡󰁲󰁥󰁳, 󰁹 󰁥󰁬 󰁲󰁥󰁳󰁴󰁯 󰁰󰁡󰁲󰁥󰁳, 󰁰󰁯󰁤󰁥󰁭󰁯󰁳 󰁧󰁥󰁮󰁥󰁲󰁡󰁲 󰁵󰁮 󰁲󰁥󰁣󰁯󰁲󰁲󰁩󰁤󰁯 󰁱󰁵󰁥 󰁥󰁭󰁰󰁩󰁥󰁺󰁡 󰁥󰁮 󰁵󰁮󰁯 󰁤󰁥 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩󰁭󰁰󰁡󰁲󰁥󰁳, 󰁹 󰁡󰁣󰁡󰁢󰁡 󰁥󰁮 󰁥󰁬 󰁯󰁴󰁲󰁯, 󰁤󰁥 󰁦󰁯󰁲󰁭󰁡 󰁱󰁵󰁥 󰁰󰁯󰁲 󰁥󰁬 󰁲󰁥󰁳󰁴󰁯 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁳󰁡󰁬󰁧󰁡 󰁥󰁬 󰁭󰁩󰁳󰁭󰁯 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁡󰁲󰁣󰁯󰁳 󰁱󰁵󰁥 󰁥󰁮󰁴󰁲󰁥, 󰁹󰁡 󰁱󰁵󰁥 󰁳󰁯󰁮  󰁰󰁡󰁲󰁥󰁳. A󰁳󰃭 󰁧󰁥󰁮󰁥󰁲󰁡󰁭󰁯󰁳 󰁵󰁮󰁡 󰁣󰁯󰁬󰁡 󰁡󰁢󰁩󰁥󰁲󰁴󰁡, 󰁡󰁢󰁩 󰁥󰁲󰁴󰁡, 󰁹 󰁣󰁯󰁮 󰁥󰁬󰁬󰁯 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁳󰁥󰁭󰁩󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯

4.󰁇󰁒󰁁󰁆󰁏󰁓 󰁈󰁁󰁍󰁉󰁌󰁔󰁏󰁎󰁉󰁁󰁎󰁁󰁎󰁏 E󰁮 󰁥󰁬 󰁡󰃱󰁯 1859 󰁥󰁬 󰁭󰁡󰁴󰁥󰁭󰃡󰁴󰁩󰁣󰁯 H󰁡󰁭󰁩󰁬󰁴󰁯󰁮 󰁰󰁵󰁳󰁯 󰁥󰁮 󰁣󰁩󰁲󰁣󰁵󰁬󰁡󰁣󰁩󰃳󰁮 󰁵󰁮 󰁲󰁯󰁭󰁰󰁥󰁣󰁡󰁢󰁥󰁺󰁡󰁳  󰁢󰁡󰁳󰁡󰁤󰁯 󰁥󰁮 󰁤󰁯󰁤󰁥󰁣󰁡󰁥󰁤󰁲󰁯 󰁲󰁥󰁧󰁵󰁬󰁡󰁲 (󰁳󰃳󰁬󰁩󰁤󰁯 󰁦󰁯󰁲󰁭󰁡󰁤󰁯 󰁰󰁯󰁲 12 󰁰󰁥󰁮󰁴󰃡󰁧󰁯󰁮󰁯󰁳 󰁩󰁧󰁵󰁡󰁬󰁥󰁳). E󰁬 󰁪󰁵󰁥󰁧󰁯 󰁣󰁯󰁮󰁳󰁩󰁳󰁴󰃭󰁡 󰁥󰁮 󰁶󰁥󰁲 󰁳󰁩 󰁳󰁥 󰁥󰁲󰁡 󰁣󰁡󰁰󰁡󰁺 󰁤󰁥 󰁲󰁥󰁣󰁯󰁲󰁲󰁥󰁲 󰁬󰁯󰁳 20 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁳󰁩󰁮 󰁲󰁥󰁰󰁥󰁴󰁩󰁲 󰁮󰁩󰁮󰁧󰁵󰁮󰁯, 󰁳󰁡󰁬󰁶󰁯 󰁥󰁬 󰁩󰁮󰁩󰁣󰁩󰁡󰁬 󰁹 󰁥󰁬 󰁦󰁩󰁮󰁡󰁬 󰁱󰁵󰁥󰁱󰁵󰁥 󰁥󰁲󰁡󰁮󰁰󰁡󰁳󰁡󰁢󰁡󰁮 󰁬󰁯󰁳 󰁭󰁩󰁳󰁭󰁯󰁳, 󰁭󰁯󰁶󰁩󰃩󰁮󰁤󰁯󰁳󰁥 󰁰󰁯󰁲 󰁬󰁡󰁳 E󰁬 󰁪󰁵󰁥󰁧󰁯 󰁳󰁯󰁬󰁵󰁣󰁩󰃳󰁮, 󰁨󰁡󰁢󰃭󰁡 󰁶󰁡󰁲󰁩󰁯󰁳 󰁣󰁩󰁣󰁬󰁯󰁳 󰁰󰁯󰁲 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁵󰁮󰁡󰁡󰁲󰁩󰁳󰁴󰁡󰁳. 󰁳󰁯󰁬󰁡 󰁶󰁥󰁺 󰁳󰁡󰁬󰁶󰁯󰁴󰁥󰁮󰁩󰁡 󰁥󰁬 󰁰󰁲󰁩󰁭󰁥󰁲󰁯 󰁹 󰁥󰁬󰁹 󰁵󰁬󰁴󰁩󰁭󰁯. O󰁴󰁲󰁯 󰁪󰁵󰁥󰁧󰁯 󰁳󰁩󰁭󰁩󰁬󰁡󰁲 󰁰󰁲󰁯󰁰󰁵󰁥󰁳󰁴󰁯 󰁰󰁯󰁲 H󰁡󰁭󰁩󰁬󰁴󰁯󰁮 󰁥󰁲󰁡 󰁥󰁬 󰁤󰁥 󰁲󰁥󰁣󰁯󰁲󰁲󰁥󰁲 󰁴󰁯󰁤󰁡󰁳 󰁬󰁡󰁳 󰁣󰁡󰁳󰁩󰁬󰁬󰁡󰁳 󰁤󰁥󰁬 󰁡󰁪󰁥󰁤󰁲󰁥󰁺 󰁣󰁯󰁮 󰁵󰁮 󰁣󰁡󰁢󰁡󰁬󰁬󰁯 󰁳󰁩󰁮 󰁲󰁥󰁰󰁥󰁴󰁩󰁲 󰁮󰁩󰁮󰁧󰁵󰁮󰁡. E󰁮 󰁥󰁳󰁴󰁥 󰁣󰁡󰁳󰁯 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁳󰁥󰁲󰁩󰁡󰁮 󰁬󰁡󰁳 󰁣󰁡󰁳󰁩󰁬󰁬󰁡󰁳 󰁤󰁥󰁬 󰁡󰁪󰁥󰁤󰁲󰁥󰁺 󰁹 󰁬󰁡󰁳 󰁡󰁲󰁩󰁳󰁴󰁡󰁳 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁰󰁯󰁳󰁩󰁢󰁬󰁥󰁳 󰁭󰁯󰁶󰁩󰁭󰁩󰁥󰁮󰁴󰁯󰁳 󰁤󰁥󰁬 󰁣󰁡󰁢󰁡󰁬󰁬󰁯. L󰁯󰁳 󰁤󰁯󰁳 󰁰󰁲󰁯󰁢󰁬󰁥󰁭󰁡󰁳 󰁳󰁥  󰁰󰁵󰁥󰁤󰁥󰁮 󰁰󰁬󰁡󰁮󰁴󰁥󰁡󰁲 󰁡 󰁰󰁡󰁲󰁴󰁩󰁲 󰁤󰁥 󰁬󰁡 󰁤󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮 󰁤󰁥 󰁤 󰁥 󰁧󰁲󰁡󰁦󰁯 󰁨󰁡󰁭󰁩󰁬󰁴󰁯󰁮󰁩󰁡󰁮󰁯:

󰁧󰁲󰁡󰁦󰁯 󰁨󰁡󰁭󰁩󰁬󰁴󰁯󰁮󰁩󰁡󰁮󰁯 󰁡 󰁴󰁯󰁤󰁯 󰁡󰁱󰁵󰁥󰁬 󰁥󰁮 󰁥󰁬 󰁱󰁵󰁥 󰁥󰁸󰁩󰁳󰁴󰁥 󰁵󰁮 󰁣󰁡󰁭󰁩󰁮󰁯 󰁣󰁥󰁲󰁲󰁡󰁤󰁯 󰁱󰁵󰁥 󰁣󰁯󰁮󰁴󰁩󰁥󰁮󰁥 󰁴󰁯󰁤󰁯󰁳 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁵󰁮󰁡 󰁳󰁯󰁬󰁡 󰁶󰁥󰁺 ( 󰁮󰁯 󰁴󰁩󰁥󰁮󰁥 󰁰󰁯󰁲 󰁱󰁵󰁥 󰁣󰁯󰁮󰁴󰁥󰁮󰁥󰁲 󰁴󰁯󰁤󰁡󰁳 󰁬󰁡󰁳 󰁡󰁲󰁩󰁳󰁴󰁡󰁳 ). U󰁮 󰁧󰁲󰁡󰁦󰁯 󰁱󰁵󰁥 󰁰󰁯󰁳󰁥󰁡 󰁵󰁮󰁡 󰁴󰁲󰁡󰁹󰁥󰁣󰁴󰁯󰁲󰁩󰁡 󰁱󰁵󰁥 󰁰󰁡󰁳󰁥 󰁡 󰁴󰁲󰁡󰁶󰃩󰁳 󰁤󰁥 󰁣󰁡󰁤󰁡 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁥󰁳 󰁤󰁥󰁮󰁯󰁭󰁩󰁮󰁡󰁤󰁯 󰁳󰁥󰁭󰁩󰁨󰁡󰁬󰁭󰁩󰁮󰁴󰁯󰁮󰁩󰁡󰁮󰁯   󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁳󰁥 󰁤󰁥󰁮󰁯󰁭󰁩󰁮󰁡

E󰁳 󰁥󰁶󰁩󰁤󰁥󰁮󰁴󰁥 󰁱󰁵󰁥 󰁡󰁬 󰁩󰁧󰁵󰁡󰁬 󰁱󰁵󰁥 󰁯󰁣󰁵󰁲󰁲󰁥 󰁣󰁯󰁮 󰁬󰁯󰁳 E󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯󰁳, 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁨󰁡󰁭󰁩󰁬󰁴󰁯󰁮󰁩󰁡󰁮󰁯󰁳 󰁤󰁥󰁢󰁥󰁮 󰁳󰁥󰁲 󰁣󰁯󰁮󰁥󰁸󰁯󰁳. S󰁩󰁮 󰁥󰁭󰁢󰁡󰁲󰁧󰁯 󰁮󰁯 󰁳󰁥 󰁨󰁡 󰁣󰁯󰁮󰁳󰁥󰁧󰁵󰁩󰁤󰁯 󰁵󰁮󰁡 󰁰󰁲󰁯󰁰󰁩󰁥󰁤󰁡󰁤 󰁳󰁥󰁮󰁣󰁩󰁬󰁬󰁡 󰁰󰁡󰁲󰁡 󰁣󰁡󰁲󰁡󰁣󰁴󰁥󰁲󰁩󰁺󰁡󰁲 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁨󰁡󰁭󰁩󰁬󰁴󰁯󰁮󰁩󰁡󰁮󰁯󰁳 󰁣󰁯󰁭󰁯 󰁯󰁣󰁵󰁲󰁲󰃭󰁡 󰁥󰁮 󰁬󰁯 󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯󰁳. S󰁩󰁮 󰁥󰁭󰁢󰁡󰁲󰁧󰁯 󰁰󰁵󰁥󰁤󰁥 󰁣󰁯󰁭󰁰󰁲󰁯󰁢󰁡󰁲󰁳󰁥 󰁱󰁵󰁥 󰁨󰁡󰁹 󰁧󰁲󰁡󰁦󰁯󰁳 󰁥󰁵󰁬󰁥󰁲󰁩󰁡󰁮󰁯󰁳 󰁱󰁵󰁥 󰁮󰁯 󰁳󰁯󰁮 󰁨󰁡󰁭󰁩󰁬󰁴󰁯󰁮󰁩󰁡󰁮󰁯󰁳 󰁹 󰁶󰁩󰁣󰁥󰁶󰁥󰁲󰁳󰁡.   󰁅󰁪󰁥󰁭󰁰󰁬󰁯󰁳

 N󰁯 H󰁡󰁭󰁩󰁬󰁴󰁯󰁮󰁩󰁡󰁮󰁯

󰁳󰁥󰁭󰁩󰁨󰁡󰁬󰁭󰁩󰁮󰁴󰁯󰁮󰁩󰁡󰁮󰁯

H󰁡󰁭󰁩󰁬󰁴󰁯󰁮󰁩󰁡󰁮󰁯

13

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

  5.󰀭 󰁇󰁒󰁁󰁆󰁏󰁓 󰁐󰁌󰁁󰁎󰁏󰁓 P󰁡󰁲󰁡 󰁩󰁮󰁴󰁲󰁯󰁤󰁵󰁣󰁩󰁲 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁰󰁬󰁡󰁮󰁯󰁳 󰁥󰁭󰁰󰁥󰁺󰁡󰁲󰁥󰁭󰁯󰁳 󰁣󰁯󰁮 󰁵󰁮 󰁰󰁲󰁯󰁢󰁬󰁥󰁭󰁡: S󰁵󰁰󰁯󰁮󰁥󰁲 󰁱󰁵󰁥 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 󰁱󰁵󰁥 󰁵󰁮󰁩󰁲 󰁴󰁲󰁥󰁳 󰁧󰁲󰁡󰁮󰁪󰁡󰁳 (󰁡,󰁢,󰁣) 󰁹 󰁴󰁲󰁥󰁳 󰁰󰁯󰁺󰁯󰁳 󰁤󰁥 󰁡󰁧󰁵󰁡 (1,2,3), 󰁰󰁥󰁲󰁯 󰁣󰁯󰁭󰁯 󰁬󰁯󰁳 󰁧󰁲󰁡󰁮󰁪󰁥󰁲󰁯󰁳 󰁥󰁮󰁥󰁭󰁩󰁳󰁴󰁡󰁤󰁯󰁳, 󰁬󰁯󰁳 󰁣󰁡󰁭󰁩󰁮󰁯󰁳 󰁮󰁯 󰁳󰁥 󰁰󰁵󰁥󰁤󰁥󰁮 󰁣󰁲󰁵󰁺󰁡󰁲. E󰁳󰁴󰁥 󰁰󰁲󰁯󰁢󰁬󰁥󰁭󰁡 󰁣󰁯󰁮󰁳󰁩󰁳󰁴󰁥 󰁥󰁮 󰁲󰁥󰁰󰁲󰁥󰁳󰁥󰁮󰁴󰁡󰁲 󰁥󰁬 󰁧󰁲󰁡󰁦󰁯 V=󰁻󰁡,󰁢,󰁣,1,2,3󰁽 󰁣󰁯󰁮 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 G=󰁻󰁻󰁡,1󰁽󰁻󰁢,2󰁽󰁻󰁣,3󰁽󰁻󰁡,2󰁽󰁻󰁢,2󰁽󰁻󰁣,2󰁽󰁻󰁡,3󰁽󰁻󰁢,3󰁽󰁻󰁣,3󰁽󰁻󰁡,4󰁽󰁻󰁢,4󰁽󰁻󰁣,4󰁽󰁽 󰁤󰁥 󰁦󰁯󰁲󰁭󰁡 󰁱󰁵󰁥 󰁮󰁯󰁳 󰁥 󰁣󰁯󰁲󰁴󰁥󰁮 󰁬󰁯󰁳 󰁥󰁪󰁥󰁳, 󰁴󰁡󰁮 󰁳󰁯󰁬󰁯 󰁥󰁮 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳. S󰁥 󰁰󰁵󰁥󰁤󰁥 󰁣󰁯󰁭󰁰󰁲󰁯󰁢󰁡󰁲 󰁱󰁵󰁥 󰁥󰁳󰁴󰁯 󰁥󰁳 󰁩󰁭󰁰󰁯󰁳󰁩󰁢󰁬󰁥, 󰁬󰁯 󰁱󰁵󰁥 󰁮󰁯󰁳 󰁬󰁬󰁥󰁶󰁡 󰁡 󰁬󰁡 󰁤󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮 󰁳󰁩󰁧󰁵󰁩󰁥󰁮󰁴󰁥:

󰁧󰁲󰁡󰁦󰁯 󰁰󰁬󰁡󰁮󰁯 󰁥󰁳 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁤󰁯󰁮󰁤󰁥 󰁳󰁥 󰁰󰁵󰁥󰁤󰁥 󰁲󰁥󰁰󰁲󰁥󰁳󰁥󰁮󰁴󰁡󰁲 󰁥󰁮 󰁥󰁬 󰁰󰁬󰁡󰁮󰁯 󰁤󰁥 󰁦󰁯󰁲󰁭󰁡 󰁱󰁵󰁥 󰁬󰁯󰁳 󰁣󰁯󰁲󰁴󰁥󰁳 󰁥󰁮󰁴󰁲󰁥 󰁬󰁯󰁳 󰁥󰁪󰁥󰁳 󰁣󰁯󰁩󰁮󰁣󰁩󰁤󰁡󰁮 󰁣󰁯󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁤󰁥󰁬 󰁭󰁩󰁳󰁭󰁯. S󰁩 󰁥󰁸󰁩󰁳󰁴󰁥 󰁤󰁩󰁣󰁨󰁡 󰁲󰁥󰁰󰁲󰁥󰁳󰁥󰁮󰁴󰁡󰁣󰁩󰃳󰁮 󰁬󰁡 󰁬󰁬󰁡󰁭󰁡󰁲󰁥󰁭󰁯󰁳 󰁭󰁡󰁰󰁡.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: U󰁮

E󰁳󰁴󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁳󰁯󰁮 󰁩󰁭󰁰󰁯󰁲󰁴󰁡󰁮󰁴󰁥󰁳 󰁡 󰁬󰁡 󰁨󰁯󰁲󰁡 󰁤󰁥 󰁤󰁩󰁳󰁥󰃱󰁡󰁲 󰁣󰁩󰁲󰁣󰁵󰁩󰁴󰁯󰁳 󰁥󰁬󰁥󰁣󰁴󰁲󰃳󰁮󰁩󰁣󰁯󰁳, 󰁹󰁡 󰁱󰁵󰁥 󰁳󰁩 󰁤󰁯󰁳 󰁣󰁯󰁮󰁥󰁸󰁩󰁯󰁮󰁥󰁳 󰁳󰁥 󰁣󰁲󰁵󰁺󰁡󰁮 󰁰󰁲󰁯󰁤󰁵󰁣󰁥󰁮 󰁣󰁯󰁲󰁴󰁯󰁣󰁩󰁲󰁣󰁵󰁩󰁴󰁯. S󰁥 󰁰󰁵󰁥󰁤󰁥 󰁣󰁯󰁭󰁰󰁲󰁯󰁢󰁡󰁲 󰁱󰁵󰁥 󰁬󰁯󰁳 5 󰁳󰃳󰁬󰁩󰁤󰁯󰁳  󰁰󰁬󰁡󰁴󰃳󰁮󰁩󰁣󰁯󰁳, 󰁳󰁯󰁮 󰁰󰁬󰁡󰁮󰁯󰁳. E󰁳 󰁢󰁡󰁳󰁴󰁡󰁮󰁴󰁥 󰁣󰁯󰁭󰁰󰁬󰁩󰁣󰁡󰁤󰁯 󰁤󰁥󰁴󰁥󰁲󰁭󰁩󰁮󰁡󰁲 󰁳󰁩 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁥󰁳 󰁰󰁬󰁡󰁮󰁯 󰁯 󰁮󰁯. E󰁬 󰁴󰁥󰁯󰁲󰁥󰁭󰁡 󰁭󰁡󰁳 󰁳󰁥󰁮󰁣󰁩󰁬󰁬󰁯 󰁥󰁳 󰁥󰁬 󰁤󰁥 K󰁵󰁲󰁡󰁴󰁯󰁷󰁳󰁫󰁩, 󰁱󰁵󰁥 󰁤󰁩󰁣󰁥 󰁱󰁵󰁥 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁥󰁳 󰁰󰁬󰁡󰁮󰁯 󰁳󰁩 󰁹 󰁳󰁯󰁬󰁯 󰁳󰁩, 󰁮󰁯 󰁣󰁯󰁮󰁴󰁩󰁥󰁮󰁥 󰁧󰁲󰁡󰁦󰁯 󰁰󰁥󰁮󰁴󰁡󰁧󰁯󰁮󰁡󰁬󰁥󰁳 󰁯 󰁨󰁥󰁸󰁡󰁧󰁯󰁮󰁡󰁬 󰁴󰁲󰁡󰁳 󰁥󰁬󰁩󰁭󰁩󰁮󰁡󰁲 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁳󰁵󰁰󰁥󰁲󰁦󰁬󰁵󰁯󰁳 󰁥󰁸󰁩󰁳󰁴󰁥󰁮󰁴󰁥󰁳 󰁥󰁮 󰁥󰁬 󰁭󰁥󰁤󰁩󰁯 󰁤󰁥 󰁬󰁯󰁳 󰁥󰁪󰁥󰁳. L󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁰󰁬󰁡󰁮󰁯󰁳 󰁣󰁯󰁮󰁥󰁸󰁯󰁳 󰁤󰁥󰁳󰁣󰁯󰁭󰁰󰁯󰁮󰁥󰁮 󰁥󰁬 󰁰󰁬󰁡󰁮󰁯 󰁥󰁮 󰁴󰁲󰁯󰁺󰁯󰁳 󰁰󰁯󰁬󰁩󰁧󰁯󰁮󰁡󰁬󰁥󰁳, 󰁱󰁵󰁥 󰁬󰁬󰁡󰁭󰁡󰁲󰁥󰁭󰁯󰁳 󰁣󰁡󰁲󰁡󰁳. L󰁡 󰁰󰁡󰁲󰁴󰁥 󰁥󰁸󰁴󰁥󰁲󰁮󰁡 󰁤󰁥󰁬 󰁧󰁲󰁡󰁦󰁯 󰁳󰁵󰁥󰁬󰁥 󰁣󰁯󰁮󰁳󰁩󰁤󰁥󰁲󰁡󰁲󰁳󰁥 󰁴󰁡󰁭󰁢󰁩󰃩󰁮 󰁵󰁮󰁡 󰁣󰁡󰁲󰁡 󰁤󰁥 󰁥󰁳󰁴󰁥.  󰁐󰁲󰁯󰁰󰁯󰁳󰁩󰁣󰁩󰃳󰁮: E󰁮 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁰󰁬󰁡󰁮󰁯 󰁣󰁯󰁮󰁥󰁸󰁯 󰁥󰁬 󰁮󰁵󰁭󰁥󰁲󰁯 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁯(G), 󰁥󰁬 󰁮󰁵󰁭󰁥󰁲󰁯 󰁤󰁥

󰁥󰁪󰁥󰁳 󰁴(G) 󰁹 󰁥󰁬 󰁮󰁵󰁭󰁥󰁲󰁯 󰁤󰁥 󰁣󰁡󰁲󰁡󰁳 󰁣(G) 󰁣󰁵󰁭󰁰󰁬󰁥󰁮 󰁬󰁡 󰁳󰁩󰁧󰁵󰁩󰁥󰁮󰁴󰁥 󰁲󰁥󰁬󰁡󰁣󰁩󰃳󰁮: 󰁯(󰁇)+󰁣(󰁇)󰀽󰁴(󰁇)+󰀲󰀮  󰁄󰁥󰁭󰁯󰁳󰁴󰁲󰁡󰁣󰁩󰃳󰁮: 󰁬󰁡 󰁨󰁡󰁲󰁥󰁭󰁯󰁳 󰁰󰁯󰁲 󰁩󰁮󰁤󰁵󰁣󰁣󰁩󰃳󰁮 󰁳󰁯󰁢󰁲󰁥 󰁥󰁬 󰁮󰁵󰁭󰁥󰁲󰁯 󰁤󰁥 󰁣󰁡󰁲󰁡󰁳.

󰁡)  󰁥󰁬 󰁧󰁲󰁡󰁦󰁯 󰁴󰁩󰁥󰁮󰁥 󰁵󰁮󰁡 󰁳󰁯󰁬󰁡 󰁣󰁡󰁲󰁡, 󰁬󰁵󰁥󰁧󰁯 󰁮󰁯 󰁴󰁩󰁥󰁮󰁥 󰁣󰁩󰁣󰁬󰁯󰁳 󰁣󰁯󰁮 󰁬󰁯 󰁱󰁵󰁥 󰁯(G)=󰁴(󰁧)+1   󰁯(G)+󰁣(G)=󰁴(G)+2, 󰁹󰁡 󰁱󰁵󰁥 󰁣(G)=1.  󰁢)  S󰁵󰁰󰁯󰁮󰁥󰁲 󰁶󰁡󰁬󰁩󰁤󰁡 󰁬󰁡 󰁲󰁥󰁧󰁬󰁡 󰁰󰁡󰁲󰁡 󰁫󰀭1 󰁣󰁡󰁲󰁡󰁳. 󰁯(G󰁫󰀭1)+󰁫󰀭1=󰁴(G󰁫󰀭1)+2 󰁣)  S󰁩 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁣󰁯 󰁫 󰁣󰁡󰁲󰁡󰁳, 󰁰󰁯󰁤󰁥󰁭󰁯󰁳 󰁣󰁯󰁮󰁳󰁴󰁲󰁵󰁩󰁲󰁬󰁯 󰁡󰃱󰁡󰁤󰁩󰁥󰁮󰁤󰁯 󰁬󰁡 󰁣󰁡󰁲󰁡 󰁥󰁸󰁴󰁥󰁲󰁩󰁯󰁲 󰁡 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁣󰁯󰁮 󰁫󰀭1 󰁣󰁡󰁲󰁡󰁳. T󰁥󰁮󰁥󰁭󰁯󰁳 󰁱󰁵󰁥 󰁳󰁩 󰁬󰁡 󰁣󰁡󰁲󰁡 󰁴󰁩󰁥󰁮󰁥 󰁲 󰁶󰃩󰁴󰁩󰁣󰁥󰁳, 󰁴󰁥󰁮󰁭󰁯󰁳 󰁱󰁵󰁥 󰁨󰁡󰁹 󰁲+1 󰁥󰁪󰁥󰁳 󰁮󰁵󰁥󰁶󰁯󰁳, 󰁹󰁡 󰁱󰁵󰁥 󰁴󰁥󰁮󰁥󰁭󰁯 󰁱󰁵󰁥 󰁵󰁮󰁩󰁲 󰁬󰁯󰁳 󰁲 󰁶󰁥󰁲󰁴󰁩󰁣󰁥󰁳 󰁥󰁮󰁴󰁲󰁥 󰁳󰁩 (󰁲󰀭1 󰁥󰁪󰁥󰁳) 󰁹 󰁤󰁯󰁳 󰁤󰁥 󰁥󰁬󰁬󰁯󰁳 󰁣󰁯󰁮 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁡󰁮󰁴󰁩󰁧󰁵󰁯󰁳. A󰁳󰃭 󰁥󰁬 󰁮󰁵󰁥󰁶󰁯 󰁧󰁲󰁡󰁦󰁯 󰁴󰁥󰁮󰁤󰁲󰁡 󰁫 󰁣󰁡󰁲󰁡󰁳, 󰁯(G󰁫󰀭1) +󰁲 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁹 󰁴(G󰁫󰀭1)+󰁲+1 󰁥󰁪󰁥󰁳. 󰁯(G󰁫)+󰁫=󰁯(G󰁫󰀭1)+󰁲+󰁫; 󰁴(G󰁫)=󰁴(G󰁫󰀭1)+󰁲+1  󰁯(G󰁫󰀭1)+󰁲+󰁫󰀭1+1 󰂿=? 󰁴(G󰁫󰀭1)+󰁲+1+󰁲+1=󰁲+12 C󰁯󰁭󰁯 󰁰󰁯󰁲 󰁨󰁩󰁰󰁯󰁴󰁥󰁳󰁩󰁳 󰁤󰁥 󰁩󰁮󰁤󰁵󰁣󰁣󰁩󰃳󰁮 󰁯(G󰁫󰀭1)+󰁫󰀭1=󰁴(G󰁫󰀭1)+2, 󰁥󰁮󰁴󰁯󰁮󰁣󰁥󰁳 󰁬󰁡 󰁦󰁯󰁲󰁭󰁵󰁬󰁡 󰁡󰁮󰁴󰁥󰁲󰁩󰁯󰁲 󰁱󰁵󰁥󰁤󰁡 󰁲+1=󰁲+1. L󰁵󰁥󰁧󰁯 󰁥󰁳 󰁣󰁩󰁥󰁲󰁴󰁯

14

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

  6.󰀭󰁄󰁉󰁁󰁇󰁒󰁁󰁍󰁁󰁓 󰁄󰁅 󰁁󰁒󰁂󰁏󰁌 U󰁮 󰁣󰁡󰁳󰁯 󰁥󰁳󰁰󰁥󰁣󰁩󰁡󰁬 󰁥󰁮 󰁬󰁯󰁳 󰁧󰁲󰁡󰁦󰁯󰁳 󰁳󰁯󰁮 󰁬󰁡󰁳 󰁥󰁳󰁴󰁲󰁵󰁣󰁴󰁵󰁲󰁡󰁳 󰁤󰁥󰁮󰁯󰁭󰁩󰁮󰁡󰁤󰁯󰁳 󰁤󰁩󰁡󰁧󰁲󰁡󰁭󰁡󰁳 󰁤󰁥 󰃡󰁲󰁢󰁯󰁬, 󰁩󰁮󰁴󰁲󰁯󰁤󰁵󰁣󰁩󰁲 󰁰󰁯󰁲 󰁰󰁲󰁩󰁭󰁥󰁲󰁡 󰁶󰁥󰁺 󰁥󰁮 󰁬󰁯󰁳 󰁴󰁲󰁡󰁢󰁡󰁪󰁯󰁳 󰁤󰁥 K󰁩󰁲󰁣󰁨󰁯󰁦󰁦 (1847) 󰁤󰁥 󰁲󰁥󰁤󰁥󰁳 󰁥󰁬󰃩󰁣󰁴󰁲󰁩󰁣󰁡󰁳.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: U󰁮 󰃡󰁲󰁢󰁯󰁬  󰁥󰁳 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁣󰁯󰁮󰁥󰁸󰁯 󰁳󰁩󰁮 󰁣󰁩󰁣󰁬󰁯󰁳.  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: 󰁂󰁯󰁳󰁱󰁵󰁥 󰁥󰁳 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 󰁮󰁯 󰁣󰁯󰁮󰁥󰁸󰁯 󰁳󰁩󰁮 󰁣󰁩󰁣󰁬󰁯 (󰁵󰁮󰁩󰃳󰁮 󰁶󰁡󰁲󰁩󰁯󰁳 󰃡󰁲󰁢󰁯󰁬󰁥󰁳). C󰁯󰁭󰁯 󰁮󰁯 󰁨󰁡󰁹 󰁣󰁩󰁣󰁬󰁯 󰁥󰁮 󰁥󰁬 󰃡󰁲󰁢󰁯󰁬, 󰁵󰁮󰁡 󰁶󰁥󰁺 󰁱󰁵󰁥 2 󰁣󰁡󰁤󰁥󰁮󰁡󰁳 󰁳󰁥 󰁳󰁥󰁰󰁡󰁲󰁡󰁮 󰁮󰁯 󰁶󰁵󰁥󰁬󰁶󰁥󰁮 󰁹󰁡 󰁡  󰁪󰁵󰁮󰁴󰁡󰁲󰁳󰁥, 󰁡󰁬 󰁩󰁧󰁵󰁡󰁬 󰁱󰁵󰁥 󰁯󰁣󰁵󰁲󰁲󰁥󰁮 󰁥󰁮 󰁬󰁡󰁳 󰁲󰁡󰁭󰁡󰁳 󰁤󰁥 󰁵󰁮 󰃡󰁲󰁢󰁯󰁬 󰁡󰁬 󰁳󰁥󰁰󰁡󰁲󰁡󰁲󰁳󰁥 󰁤󰁥󰁬 󰁴󰁲󰁯󰁮󰁣󰁯 󰁯 󰁤󰁥 󰁲󰁡󰁭󰁡󰁳 󰁡󰁮󰁴󰁥󰁲󰁩󰁯󰁲󰁥󰁳. G󰁥󰁮󰁥󰁲󰁡󰁬󰁭󰁥󰁮󰁴󰁥 󰁰󰁡󰁲󰁡 󰁲󰁥󰁰󰁲󰁥󰁳󰁥󰁮󰁴󰁡󰁲 󰁵󰁮 󰃡󰁲󰁢󰁯󰁬 󰁳󰁥 󰁥󰁬󰁩󰁧󰁥 󰁵󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁣󰁵󰁡󰁬󰁥󰁳󰁱󰁵󰁩󰁥󰁲󰁡(󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁲󰁡󰃭󰁺), 󰁡 󰁣󰁯󰁮󰁴󰁩󰁮󰁵󰁡󰁣󰁩󰃳󰁮 󰁳󰁥 󰁲󰁥󰁰󰁲󰁥󰁳󰁥󰁮󰁴󰁡󰁮 󰁬󰁯󰁳 󰁱󰁵󰁥 󰁥󰁳󰁴󰃡󰁮 󰁤󰁩󰁲󰁥󰁣󰁴󰁡󰁭󰁥󰁮󰁴󰁥 󰁵󰁮󰁩󰁤󰁯󰁳 󰁡 󰁥󰁬 󰁰󰁯󰁲 󰁵󰁮 󰁥󰁪󰁥 󰁤󰁥󰁳󰁣󰁥󰁮󰁤󰁥󰁮󰁴󰁥, 󰁣󰁯󰁮󰁳󰁩󰁧󰁵󰁩󰁥󰁮󰁤󰁯 󰁡󰁳󰃭 󰁱󰁵󰁥 󰁥󰁳󰁴󰃩󰁮 󰁵󰁮󰁩󰁤󰁯󰁳 󰁨󰁡󰁳󰁴󰁡 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁦󰁩󰁮󰁡󰁬󰁥󰁳(󰁳󰁩󰁮 󰁤󰁥󰁳󰁣󰁥󰁮󰁤󰁩󰁥󰁮󰁴󰁥󰁳). (1) 1

2

3

4

2

(1)=(2) , 󰁰󰁥󰁲󰁯 󰁥󰁬 󰃡󰁲󰁢󰁯󰁬 󰁳󰁥 󰁳󰁵󰁥󰁬󰁥 󰁲󰁥󰁰󰁲󰁥󰁳󰁥󰁮󰁴󰁡󰁲 󰁣󰁯󰁭󰁯 (2)

(2) 1 4

3

C󰁯󰁮󰁯󰁣󰁩󰁥󰁮󰁤󰁯 󰁥󰁬 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁲󰁡󰃭󰁺 󰁹 󰁬󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󲀜󰁨󰁩󰁪󰁯󰁳 󰁤󰁥 󲀜 󰁤󰁥 󰁣󰁡󰁤󰁡 󰁶󰃩󰁲󰁴󰁩󰁣󰁥, 󰁱󰁵󰁥 󰁮󰁯󰁳 󰁬󰁬󰁥󰁶󰁡 󰁡 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁱󰁵󰁥 󰁥󰁳󰁴󰃡󰁮 󰁵󰁮󰁩󰁤󰁯󰁳 󰁡 󰃩󰁳󰁴󰁥 󰁥󰁮 󰁳󰁥󰁮󰁴󰁩󰁤󰁯 󰁤󰁥󰁳󰁣󰁥󰁮󰁤󰁥󰁮󰁴󰁥, 󰁣󰁯󰁮󰁯󰁣󰁥󰁭󰁯󰁳 󰁥󰁬 󰃡󰁲󰁢󰁯󰁬 󰁥󰁮󰁴󰁥󰁲󰁯. H󰁩󰁪󰁯󰁳: V 1  4 

V*=V󰀭󰁻V󰃩󰁲󰁴󰁩󰁣󰁥 󰁲󰁡󰃭󰁺󰁽 2,4 3

 󰁐󰁲󰁯󰁰󰁯󰁳󰁩󰁣󰁩󰃳󰁮 󰀱: U󰁮 󰃡󰁲󰁢󰁯󰁬 󰁣󰁯󰁮 󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁴󰁩󰁥󰁮󰁥󰁮 󰁮󰀭1 󰁥󰁪󰁥󰁳.  󰁄󰁥󰁭󰁯󰁳󰁴󰁲󰁡󰁣󰁩󰃳󰁮: C󰁯󰁬󰁯󰁣󰁡󰁮󰁤󰁯 󰁥󰁬 󰃡󰁲󰁢󰁯󰁬 󰁥󰁮 󰁦󰁯󰁲󰁭󰁡 󰁰󰁡󰁤󰁲󰁥󰀭󰁤󰁥󰁳󰁣󰁥󰁮󰁤󰁩󰁥󰁮󰁴󰁥, 󰁳󰁥 󰁯󰁢󰁳󰁥󰁲󰁶󰁡 󰁱󰁵󰁥

󰁴󰁯󰁤󰁯 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁴󰁩󰁥󰁮󰁥 󰁵󰁮 󰁰󰁡󰁤󰁲󰁥, 󰁭󰁥󰁮󰁯󰁳 󰁥󰁬 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁲󰁡󰃭󰁺 󰁱󰁵󰁥 󰁮󰁯 󰁴󰁩󰁥󰁮󰁥 󰁡󰁳󰁣󰁥󰁮󰁤󰁥󰁮󰁴󰁥. L󰁵󰁥󰁧󰁯 󰁬󰁡 󰁡󰁰󰁬󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁰󰁡󰁤󰁲󰁥 󰁤󰁥 󰁥󰁳 󰁵󰁮 󰁢󰁩󰁹󰁥󰁣󰁩󰃳󰁮 󰁥󰁮󰁴󰁲󰁥 󰁥󰁪󰁥󰁳 󰁹 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁥󰁸󰁣󰁬󰁵󰁹󰁥󰁮󰁤󰁯 󰁡󰁬 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁲󰁡󰁩󰁺. L󰁵󰁥󰁧󰁯 󰁣󰁡󰁤󰁡 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁴󰁩󰁥󰁮󰁥 󰁡󰁳󰁯󰁣󰁩󰁡󰁤󰁯 󰁵󰁮 󰁥󰁪󰁥, 󰁭󰁥󰁮󰁯󰁳 󰁥󰁬 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁲󰁡󰃭󰁺, 󰁬󰁵󰁥󰁧󰁯 󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳  󰁮󰀭1 󰁥󰁪󰁥󰁳.

 󰁐󰁲󰁯󰁰󰁯󰁳󰁩󰁣󰁩󰃳󰁮 󰀲: 󰁵󰁮 󰁢󰁯󰁳󰁱󰁵󰁥 󰁣󰁯󰁮 󰁫 󰁣󰁯󰁭󰁰󰁯󰁮󰁥󰁮󰁴󰁥󰁳 󰁣󰁯󰁮󰁥󰁸󰁡󰁳 󰁹 󰁣󰁯󰁮 󰁮 󰁶󰃩󰁴󰁩󰁣󰁥󰁳 󰁴󰁩󰁥󰁮󰁥 󰁮󰀭󰁫 󰁥󰁪󰁥󰁳.

15

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳  󰁄󰁥󰁭󰁯󰁳󰁴󰁲󰁡󰁣󰁩󰃳󰁮: C󰁯󰁭󰁯 󰁥󰁬 󰁢󰁯󰁳󰁱󰁵󰁥 󰁴󰁩󰁥󰁮󰁥 󰁫 󰃡󰁲󰁢󰁯󰁬󰁥󰁳 󰁮󰁯 󰁣󰁯󰁮󰁥󰁣󰁴󰁡󰁤󰁡󰁳: 󰁥󰁪󰁥󰁳= Σ󰁫 󰁥󰁪󰁥󰁳 󰁥󰁪󰁥󰁳 󰃡󰁲󰁢󰁯󰁬=

=Σ󰁫(󰁮󰁫 󰂴󰀭1)= 󰂴󰀭1)= Σ󰁫 󰁮󰁫 󰂴󰀭 󰂴󰀭 󰁫= 󰁮󰀭󰁫 L󰁯󰁳 󰁵󰁳󰁯󰁳 󰃡󰁲󰁢󰁯󰁬󰁥󰁳 󰁳󰁯󰁮 󰁭󰁵󰁹 󰁶󰁡󰁲󰁩󰁡󰁤󰁯󰁳: 󰁤󰁥 󰁩󰁮󰁦󰁯󰁲󰁭󰁡󰁣󰁩󰃳󰁮 •  A󰁬󰁭󰁡󰁣󰁥󰁮󰁡󰁭󰁩󰁥󰁮󰁴󰁯 󰀭  O󰁲󰁤󰁥󰁮󰁡󰁤󰁯󰁲󰁥󰁳(󰁤󰁩󰁳󰁣󰁯 󰁤󰁵󰁲󰁯) 󰀭  R󰁥󰁤󰁥󰁳 󰁯󰁲󰁤󰁥󰁮󰁡󰁤󰁯󰁲󰁥󰁳(I󰁮󰁴󰁥󰁲󰁮󰁥󰁴) 󰀭  B󰁩󰁢󰁬󰁩󰁯󰁴󰁥󰁣󰁡 •  R󰁥󰁤󰁥󰁳 󰁳󰁵󰁭󰁩󰁮󰁩󰁳󰁴󰁲󰁯 󰁥󰁬󰃩󰁣󰁴󰁲󰁩󰁣󰁯, 󰁡󰁧󰁵󰁡... •  R󰁥󰁳󰁯󰁬󰁵󰁣󰁩󰃳󰁮 󰁤󰁥 󰁣󰁩󰁲󰁣󰁵󰁩󰁴󰁯󰁳 󰁥󰁬󰃩󰁣󰁴󰁲󰁩󰁣󰁯󰁳 •  C󰁬󰁡󰁳󰁩󰁦󰁩󰁣󰁡󰁣󰁩󰃳󰁮 󰁤󰁥 󰁥󰁳󰁰󰁥󰁣󰁩󰁥󰁳. •  P󰁲󰁯󰁧󰁲󰁡󰁭󰁡󰁣󰁩󰃳󰁮 󰁹 󰁡󰁮󰃡󰁬󰁩󰁳󰁩󰁳 󰁤󰁥 󰁡󰁬󰁧󰁯󰁲󰁩󰁴󰁭󰁯󰁳 •  R󰁥󰁤󰁵󰁣󰁣󰁩󰃳󰁮 󰁤󰁥 󰁰󰁲󰁯󰁢󰁬󰁥󰁭󰁡󰁳 󰁤󰁥 󰁰󰁲󰁯󰁢󰁡󰁢󰁩󰁬󰁩󰁤󰁡󰁤.  󰁒󰁥󰁬󰁡󰁣󰁩󰃳󰁮 󰃡󰁲󰁢󰁯󰁬󰁥󰁳 󰁹 󰁮󰁵󰁭󰁥󰁲󰁡󰁣󰁩󰃳󰁮 󰁮󰁵󰁭󰁥󰁲󰁡󰁣󰁩󰃳󰁮:

󰁰󰁯󰁲 󰁥󰁪󰁥󰁭󰁰󰁬󰁯 󰁥󰁮 󰁣󰁯󰁤󰁩󰁧󰁯 󰁤󰁥󰁣󰁩󰁭󰁡󰁬(󰁯 󰁢󰁩󰁮󰁡󰁲󰁩󰁯 󰁥󰁮 󰁬󰁯󰁳 󰁯󰁲󰁤󰁥󰁮󰁡󰁤󰁯󰁲󰁥󰁳). P󰁡󰁲󰁡 󰁧󰁵󰁡󰁲󰁤󰁡󰁲 󰁵󰁮 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 3 󰁣󰁩󰁦󰁲󰁡󰁳 󰁥󰁮 󰁣󰃳󰁤󰁩󰁧󰁯 󰁤󰁥󰁣󰁩󰁭󰁡󰁬 󰁮󰁯 󰁮󰁥󰁣󰁥󰁳󰁩󰁴󰁡󰁭󰁯󰁳 1000 󲀜󰁡󰁬󰁭󰁡󰁣󰁥󰁮󰁡󰁭󰁩󰁥󰁮󰁴󰁯 󰁤󰁥 󰁭󰁥󰁭󰁯󰁲󰁩󰁡󲀝, 󰁳󰁩󰁮󰁯 3 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯󰁳 󰁣󰁯󰁮 󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤 󰁤󰁥 󰁡󰁬󰁭󰁡󰁣󰁥󰁮󰁡󰁲 10 󰁤󰁩󰁦󰁥󰁲󰁥󰁮󰁴󰁥󰁳 󰁶󰁡󰁬󰁯󰁲󰁥󰁳(0,1,2,3,4,5,6,7,8,9), 󰁥󰁳󰁴󰁲󰁵󰁣󰁴󰁵󰁲󰁡󰁤󰁯󰁳 󰁥󰁮 󰁦󰁯󰁲󰁭󰁡 󰁤󰁥 󰃡󰁲󰁢󰁯󰁬. 0

1

2  ......

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2  3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

..... .. ................................  0 1 2 3 4 5 6 7 8 9...........................................................................................  N󰃺󰁭󰁥󰁲󰁯:225 . L󰁡 󰁯󰁲󰁤󰁥󰁮󰁡󰁣󰁩󰃳󰁮 󰁰󰁯󰁲 󰃡󰁲󰁢󰁯󰁬 󰁰󰁥󰁲󰁭󰁩󰁴󰁥 󰁤󰁥󰁣󰁯󰁤󰁩󰁦󰁩󰁣󰁡󰁲 󰁬󰁡 󰁩󰁮󰁦󰁯󰁲󰁭󰁡󰁣󰁩󰃳󰁮 󰁤󰁥 󰁦󰁯󰁲󰁭󰁡 󰁲󰃡󰁰󰁩󰁤󰁡 󰁡 󰁰󰁡󰁲󰁴󰁩󰁲 󰁤󰁥 󰁵󰁮 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁲󰁡󰃭󰁺 󰁤󰁡󰁤󰁯.. P󰁯󰁲 󰁥󰁪󰁥󰁭󰁰󰁬󰁯 󰁥󰁮 󰁵󰁮 󰃡󰁲󰁢󰁯󰁬 󰁨󰁯󰁭󰁯󰁧󰃩󰁮󰁥󰁯(󰁭󰁩󰁳󰁭󰁡󰁳 󰁲󰁡󰁭󰁡󰁳 󰁣󰁡󰁤󰁡 󰁶󰃩󰁲󰁴󰁩󰁣󰁥), 󰁣󰁯󰁭󰁯 󰁥󰁳 󰁥󰁬 󰁣󰁡󰁳󰁯 󰁡󰁮󰁴󰁥󰁲󰁩󰁯󰁲 󰁤󰁥󰁬 󰁮󰂺 󰁤󰁥 0 󰁡 100., 󰁥󰁮 󰁤󰁯󰁮󰁤󰁥 󰁡 󰁣󰁡󰁤󰁡 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁬󰁥 󰁳󰁡󰁬󰁥󰁮 10 󰁲󰁡󰁭󰁡󰁳(󰁴󰁡󰁮󰁴󰁯󰁳 󰁣󰁯󰁭󰁯 󰁣󰁩󰁦󰁲󰁡󰁳), 󰁬󰁡󰁳 󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤󰁥󰁳 󰁤󰁩󰁳󰁭󰁩󰁮󰁵󰁹󰁥󰁮 󰁤󰁥 󰁦󰁯󰁲󰁭󰁡 󰁬󰁯󰁧󰁡󰁲󰃭󰁴󰁭󰁩󰁣󰁡 󰁥󰁮 󰁢󰁡󰁳󰁥10: 1.  I󰁮󰁩󰁣󰁩󰁡󰁬󰁭󰁥󰁮󰁴󰁥 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 1000 󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤󰁥󰁳 (󰁤󰁥󰁬 0 󰁡󰁬 999) 2.  D󰁥󰁳󰁰󰁵󰃩󰁳 󰁤󰁥󰁬 󰁰󰁲󰁩󰁭󰁥󰁲 󰁰󰁡󰁳󰁯(1󰁥󰁲  󰁮󰁩󰁶󰁥󰁬), 󰁡󰁬 󰁥󰁬󰁥󰁧󰁩󰁲 󰁥󰁬 2 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 100 󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤󰁥󰁳(󰁤󰁥󰁬 200 󰁡󰁬 299). 3.  D󰁥󰁳󰁰󰁵󰃩󰁳 󰁤󰁥󰁬 󰁳󰁥󰁧󰁵󰁮󰁤󰁯 󰁰󰁡󰁳󰁯(2󰂺 󰁮󰁩󰁶󰁥󰁬), 󰁡󰁬 󰁥󰁬󰁥󰁧󰁩󰁲 󰁥󰁬 2 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 10 󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤󰁥󰁳 (󰁤󰁥󰁬 220 󰁡󰁬 229). 4.  D󰁥󰁳󰁰󰁵󰃩󰁳 󰁤󰁥󰁬 4󰂺 󰁰󰁡󰁳󰁯(3󰁥󰁲   󰁮󰁩󰁶󰁥󰁬),󰁡󰁬 󰁥󰁬󰁥󰁧󰁩󰁲 󰁥󰁬 5 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 󰁥󰁬 󰁮󰁵󰁭󰁥󰁲󰁯 󰁢󰁵󰁳󰁣󰁡󰁤󰁯(󰁵󰁮󰁡  󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤󰁥󰁳)

16

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

M󰁥󰁤󰁩󰁡󰁮󰁴󰁥 󰁵󰁮 󰁤󰁩󰁡󰁧󰁲󰁡󰁭󰁡 󰁤󰁥 󰃡󰁲󰁢󰁯󰁬 󰁨󰁥󰁭󰁯󰁳 󰁬󰁬󰁥󰁧󰁡󰁤󰁯 󰁡󰁬 󰁮󰃺󰁭󰁥󰁲󰁯 󰁥󰁮 󰁴󰁲󰁥󰁳 󰁰󰁡󰁳󰁯󰁳, 󰁤󰁩󰁳󰁭󰁩󰁮󰁵󰁹󰁥󰁮󰁤󰁯 󰁬󰁯󰁧󰁡󰁲󰃭󰁴󰁭󰁩󰁣󰁡󰁭󰁥󰁮󰁴󰁥 󰁬󰁡󰁳 󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤󰁥󰁳: 󰁰󰁡󰁳󰁯󰀽󰁬󰁯󰁧 󰀱󰀱󰀰󰀰(󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤󰁥󰁳). S󰁩 󰁣󰁯󰁤󰁩󰁦󰁩󰁣󰃡󰁲󰁡󰁭󰁯󰁳 󰁬󰁡 󰁩󰁮󰁦󰁯󰁲󰁭󰁡󰁣󰁩󰃳󰁮 󰁥󰁮 󰁣󰃳󰁤󰁩󰁧󰁯 󰁢󰁩󰁮󰁡󰁲󰁩󰁯, 󰁥󰁮 󰁦󰁯󰁲󰁭󰁡 󰁤󰁥 󰁥󰁳󰁴󰁲󰁵󰁣󰁴󰁵󰁲󰁡 󰁤󰁥 󰃡󰁲󰁢󰁯󰁬 󰁳󰁥󰁲󰃭󰁡 󰁳󰁥󰁭󰁥󰁪󰁡󰁮󰁴󰁥 󰁳󰃳󰁬󰁯 󰁱󰁵󰁥 󰁡󰁨󰁯󰁲󰁡 󰁥󰁬 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤󰁥󰁳 󰁤󰁥󰁳󰁣󰁩󰁥󰁮󰁤󰁥 󰁬󰁯󰁧󰁡󰁲󰃭󰁭󰁩󰁣󰁡󰁭󰁥󰁮󰁴󰁥 󰁰󰁥󰁲󰁯 󰁥󰁮 󰁢󰁡󰁳󰁥 2. 󰀳

󰀱󰀰󰀰󰀰

󰀱󰀰󰀰󰀰      󰁳      󰁥        󰁤      󰁡        󰁤        󰁩        󰁬        󰁩        󰁢        󰁩      󰁳      󰁯      󰁰

󰀳

       󰀩      󰁳      󰁯      󰁰        󰀨      󰁧      󰁯        󰁬

󰀲

󰀵󰀰 󰀵󰀰󰀰 󰀰

󰀲

󰀱

󰀱󰀰 󰀰

󰀰 󰀰

󰀱󰀰

󰀱

 

󰀲

󰀱 󰀳

󰀱

󰀰 󰀴

󰀰 󰀰

󰁐󰁡󰁳󰁯󰁳

󰀱

󰀲

󰀳

󰁐󰁡󰁳󰁯󰁳

E󰁮 󰃡󰁲󰁢󰁯󰁬󰁥󰁳 󰁮󰁯 󰁨󰁯󰁭󰁯󰁧󰃩󰁮󰁥󰁯󰁳, 󰁤󰁯󰁮󰁤󰁥 󰁣󰁡󰁤󰁡 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁮󰁯 󰁴󰁩󰁥󰁮󰁥 󰁥󰁬 󰁭󰁩󰁳󰁭󰁯 󰁮󰂺 󰁤󰁥 󰁲󰁡󰁭󰁡󰁳 󰁤󰁥󰁳󰁣󰁥󰁮󰁤󰁩󰁥󰁮󰁴󰁥󰁳, 󰁬󰁡 󰁧󰁲󰃡󰁦󰁩󰁣󰁡 󰁮󰁯 󰁥󰁳 󰁴󰁡󰁮 󰁳󰁩󰁭󰁰󰁬󰁥 󰁰󰁵󰁥󰁳 󰁥󰁬 󰁮󰂺 󰁤󰁥 󰁰󰁯󰁳󰁩󰁢󰁩󰁬󰁩󰁤󰁡󰁤󰁥󰁳 󰁤󰁥󰁳󰁣󰁩󰁥󰁮󰁤󰁥 󰁤󰁥 󰁤󰁩󰁦󰁥󰁲󰁥󰁮󰁴󰁥 󰁦󰁯󰁲󰁭󰁡 󰁳󰁥󰁧󰃺󰁮 󰁥󰁬 󰁣󰁡󰁭󰁩󰁮󰁯 󰁥󰁬󰁥󰁧󰁩󰁤󰁯. P󰁥󰁲󰁯 󰁥󰁮 󰁴󰁯󰁤󰁯 󰁣󰁡󰁳󰁯 󰁥󰁬 󰁤󰁥󰁣󰁲󰁥󰁣󰁩󰁭󰁩󰁥󰁮󰁴󰁯 P󰁯󰁳󰁩󰁢󰀭󰁰󰁡󰁳󰁯󰁳 󰁥󰁳 󰁧󰁲󰁡󰁮󰁤󰁥.

7.󰀭󰁍󰁁󰁔󰁒󰁉󰁃󰁅󰁓 󰁁󰁓󰁏󰁃󰁉󰁁󰁄󰁁󰁓 󰁁 󰁕󰁎 󰁇󰁒󰁁󰁆󰁏  󰁄󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮: D󰁡󰁤󰁯 󰁵󰁮 󰁧󰁲󰁡󰁦󰁯 (V,E) 󰁬󰁬󰁡󰁭󰁡󰁭󰁯󰁳 󰁭󰁡󰁴󰁲󰁩󰁺 󰁤󰁥 󰁡󰁤󰁹󰁡󰁣󰁥󰁮󰁴󰁥󰁳 󰁡 󰁬󰁡 󰁭󰁡󰁴󰁲󰁩󰁺

M∈M󰁮󰁸󰁮(N) 󰁤󰁯󰁮󰁤󰁥 󰁮 󰁳󰁯󰁮 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁤󰁥 V 󰁹 󰁤󰁥󰁦󰁩󰁮󰁩󰁤󰁡 󰁣󰁯󰁭󰁯:  1  󰁳󰁩 󰁡󰁲󰁣󰁯 (󰁩,  󰁪 ) ∈ 󰁅 (󰁵󰁮󰁩󰁤󰁯󰁳 )   󰁭󰁩󰁪 =  0  󰁳󰁩 󰁡󰁲󰁣󰁯 (󰁩,  󰁪 ) ∉ 󰁅 (󰁮󰁯 󰁵󰁮󰁩󰁤󰁯󰁳 ) L󰁡 󰁭󰁡󰁴󰁲󰁩󰁺 󰁥󰁮 󰁧󰁲󰁡󰁦󰁯󰁳 󰁮󰁯 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯󰁳 󰁥󰁳 󰁳󰁩󰁭󰃩󰁴󰁲󰁩󰁣󰁡, 󰁰󰁵󰁥󰁳 󰁴󰁯󰁤󰁯 󰁰󰁡󰁲 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 (󰁩,󰁪) 󰁵󰁮󰁩󰁤󰁯󰁳 󰁥󰁮 󰁬󰁯󰁳 󰁤󰁯󰁳 󰁳󰁥󰁮󰁴󰁩󰁤󰁯󰁳. E󰁮 󰁧󰁲󰁡󰁦󰁯󰁳 󰁯󰁲󰁩󰁥󰁮󰁴󰁡󰁤󰁯󰁳 󰁬󰁡 󰁤󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮 󰁤󰁥 󰁬󰁡 󰁭󰁡󰁴󰁲󰁩󰁺 󰁤󰁥 󰁡󰁤󰁹󰁡󰁣󰁥󰁮󰁴󰁥 󰁥󰁳 󰁬󰁡 󰁭󰁩󰁳󰁭󰁡, 󰁰󰁥󰁲󰁯 󰁡󰁨󰁯󰁲󰁡 󰁮󰁯 󰁥󰁳 󰁳󰁩󰁭󰃩󰁴󰁲󰁩󰁣󰁡.

 󰁅󰁪󰁥󰁭󰁰󰁬󰁯:

1

2

3  0 1 1    M=  1 0 1    1 1 0    

1

2

3 17

󰀴

 

 

󰁔󰁥󰁭󰁡 󰀲󰀮 󰁧󰁲󰁡󰁦󰁯󰁳

 0 1 1    M󰂴=  0 0 0       0 1 0  E󰁮 󰁥󰁬 󰁣󰁡󰁳󰁯 󰁤󰁥 󰁰󰀭󰁧󰁲󰁡󰁦󰁯󰁳 󰁨󰁡󰁹 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁱󰁵󰁥 󰁰󰁵󰁥󰁤󰁥󰁮 󰁥󰁳󰁴󰁡󰁲 󰁵󰁮󰁩󰁤󰁯󰁳 󰁰󰁯󰁲 󰁰 󰁡󰁲󰁣󰁯󰁳, 󰁬󰁵󰁥󰁧󰁯 󰁬󰁯󰁳 󰁶󰁡󰁬󰁯󰁲󰁥󰁳 󰁤󰁥 󰁭󰁩󰁪 󰁶󰁡󰁮 󰁤󰁥 0 󰁡 󰁰. 󰁔󰁥󰁯󰁲󰁥󰁭󰁡: S󰁩 󰁥󰁮 󰁬󰁡 󰁭󰁡󰁴󰁲󰁩󰁺 M 󰁯 M󰂴 󰁤󰁥󰁬 󰁧󰁲󰁡󰁦󰁯 󰁬󰁡 󰁥󰁬󰁥󰁶󰁡󰁭󰁯󰁳 󰁡󰁬 󰁣󰁵󰁡󰁤󰁲󰁡󰁤󰁯 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 󰁵󰁮󰁡

󰁭󰁡󰁴󰁲󰁩󰁺, 󰁱󰁵󰁥 󰁮󰁯󰁳 󰁩󰁮󰁤󰁩󰁣󰁡 󰁥󰁬 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁣󰁡󰁤󰁥󰁮󰁡󰁳 󰁣󰁯󰁮 2 󰁥󰁪󰁥󰁳 󰁱󰁵󰁥 󰁵󰁮󰁥󰁮 󰁤󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳. S󰁩 󰁬󰁯 󰁥󰁬󰁥󰁶󰁡󰁭󰁯󰁳 󰁡󰁬 󰁣󰁵󰁢󰁯 󰁬󰁡 󰁭󰁡󰁴󰁲󰁩󰁺 󰁲󰁥󰁳󰁵󰁬󰁴󰁡󰁤󰁯 󰁮󰁯󰁳 󰁩󰁮󰁤󰁩󰁣󰁡 󰁥󰁬 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁣󰁡󰁤󰁥󰁮󰁡󰁳 󰁣󰁯󰁮 3 󰁥󰁪󰁥󰁳 󰁱󰁵󰁥 󰁵󰁮󰁥󰁮 󰁣󰁡󰁤󰁡 󰁰󰁡󰁲 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳. A󰁳󰃭 M󰁫  󰁮󰁯󰁳 󰁩󰁮󰁦󰁯󰁲󰁭󰁡 󰁤󰁥󰁬 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁣󰁡󰁤󰁥󰁮󰁡󰁳 󰁣󰁯󰁮 󰁫 󰁥󰁪󰁥󰁳 󰁱󰁵󰁥 󰁵󰁮󰁥󰁮 󰁣󰁡󰁤󰁡  󰁰󰁡󰁲 󰁤󰁥 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳.  2 1 1     󰁍 2 =  1 2 1  1 1 2    

 2 3 3       󰁍 3 =  3 2 3   3 3 2    

 󰁄󰁥󰁭󰁯󰁳󰁴󰁲󰁡󰁣󰁩󰃳󰁮: P󰁯󰁲 󰁩󰁮󰁤󰁵󰁣󰁣󰁩󰃳󰁮 󰁰󰁯󰁤󰁥󰁭󰁯󰁳 󰁶󰁥󰁲 󰁬󰁡 󰁶󰁥󰁲󰁡󰁣󰁩󰁤󰁡󰁤 󰁤󰁥󰁬 󰁴󰁥󰁯󰁲󰁥󰁭󰁡: 1)  P󰁡󰁲󰁡 󰁫=1 󰁳󰁥 󰁣󰁵󰁭󰁰󰁬󰁥 M=1, M2,M3...=1 2)  S󰁵󰁰󰁯󰁮󰁧󰁡󰁭󰁯󰁳 󰁱󰁵󰁥 󰁰󰁡󰁲󰁡 󰁫󰀭1 󰁳󰁥 󰁣󰁵󰁭󰁰󰁬󰁥 󰁹 󰁱󰁵󰁥 A=M󰁫󰀭1 󰁣󰁵󰁭󰁰󰁬󰁥 󰁬󰁡 󰁰󰁲󰁯󰁰󰁯󰁳󰁩󰁣󰁩󰃳󰁮. 3)  E󰁬 󰁮󰂺 󰁤󰁥 󰁣󰁡󰁤󰁥󰁮󰁡󰁳 󰁣󰁯󰁮 󰁫 󰁥󰁪󰁥󰁳 󰁵󰁮󰁩󰁥󰁮󰁤󰁯 󰁩 󰁹 󰁪 󰁳󰁥󰁲󰃡 󰁬󰁡 󰁳󰁵󰁭󰁡 󰁤󰁥 󰁣󰁡󰁤󰁥󰁮󰁡󰁳 󰁣󰁯󰁮 󰁫󰀭1 󰁥󰁪󰁥󰁳

󰁵󰁮󰁩󰁥󰁮󰁤󰁯 󰁩 󰁣󰁯󰁮 󰁣󰁵󰁡󰁬󰁱󰁵󰁩󰁥󰁲 󰁶󰃩󰁲󰁴󰁩󰁣󰁥 󰁳, 󰁤󰁯󰁮󰁤󰁥 󰁥󰁸󰁩󰁳󰁴󰁡 󰁥󰁬 󰁥󰁪󰁥 (󰁳,󰁪). 󰁤󰁥󰁮󰁯󰁴󰁡󰁲󰁥󰁭󰁯󰁳 󰁥󰁬 󰁮󰂺 󰁤󰁥 󰁣󰁡󰁤󰁥󰁮󰁡󰁳 󰁱󰁵󰁥 󰁵󰁮󰁥 󰁣󰁯󰁮 󰁫󰀭1 󰁥󰁪󰁥󰁳 󰁩 󰁣󰁯󰁮 󰁳 󰁣󰁯󰁭󰁯 󰁡 󰁩󰁳.  C󰁡󰁤󰁫 (󰁩,󰁪)= (󰁩,󰁪)=   ∑ 󰁡󰁩󰁳 , 󰁣󰁯󰁭󰁯 󰁭󰁳󰁪=1 󰁳󰁩 (󰁩,󰁪)∈E 󰁹 󰁭󰁳󰁪=0 󰁳󰁩 (󰁩,󰁪)∉E, 󰁴󰁥󰁮󰁥󰁭󰁯󰁳 󰁱󰁵󰁥 󰁬󰁡 ( 󰁳 , 󰁪 )∈ 󰁅 

󰁥󰁸󰁰󰁲󰁥󰁳󰁩󰃳󰁮 󰁡󰁮󰁴󰁥󰁲󰁩󰁯󰁲 󰁳󰁥 󰁲󰁥󰁤󰁵󰁣󰁥 󰁡 󰁬󰁡 󰁳󰁩󰁧󰁵󰁩󰁥󰁮󰁴󰁥: 󰁣󰁡󰁤 󰁫  (󰁩,  󰁪 ) = ∑    󰁳  󰁡   󰁩󰁳 ⋅ 󰁭 󰁳󰁪 = 󰁢󰁩󰁪 , 󰁥󰁬󰁥󰁭󰁥󰁮󰁴󰁯 󰁩,󰁪 󰁤󰁥 󰁬󰁡 󰁭󰁡󰁴󰁲󰁩󰁺. P󰁥󰁲󰁯 󰁬󰁡 󰁭󰁡󰁴󰁲󰁩󰁺 B 󰁥󰁳 󰁬󰁡 󰁤󰁥󰁦󰁩󰁮󰁩󰁣󰁩󰃳󰁮 󰁤󰁥 󰁰󰁲󰁯󰁤󰁵󰁣󰁴󰁯 󰁥󰁮󰁴󰁲󰁥 󰁬󰁡 󰁭󰁡󰁴󰁲󰁩󰁺 A 󰁹 󰁬󰁡 M B=A •M,  󰁰󰁥󰁲󰁯 A=M󰁫󰀭1 󰁣󰁯󰁮 󰁬󰁯 󰁱󰁵󰁥 B=M󰁫 , 󰁹 󰁮󰁯󰁳 󰁤󰁥󰁦󰁩󰁮󰁥 󰁥󰁬 󰁮󰃺󰁭󰁥󰁲󰁯 󰁤󰁥 󰁣󰁡󰁤󰁥󰁮󰁡󰁳 󰁱󰁵󰁥 󰁥󰁳󰁴󰁡󰁮 󰁣󰁯󰁮󰁳󰁴󰁲󰁵󰁩󰁤󰁡󰁳 󰁣󰁯󰁮 󰁫 󰁥󰁪󰁥󰁳 󰁹 󰁱󰁵󰁥 󰁮󰁯󰁳 󰁵󰁮󰁥󰁮 󰁬󰁯󰁳 󰁶󰃩󰁲󰁴󰁩󰁣󰁥󰁳 󰁩 󰁣󰁯󰁮 󰁪.

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