Tea Gaft Teesefa

1
Te???a G?af?µ?t??Teµe???se??-???????µ??-?fa?µ????

• ?ef??a?? 4 S??desµ???t?ta

2
??sa???? (1)

• ?????a t?? s??desµ???t?ta? ??s?t?ta
  s??desµ???t?ta?

3
??sa???? (2)

• S????? ??µß?? t?µ?? V e??a? t? s????? t??
  ??µß??, t?t??? ?ste t? ???f?µa G-V ?? ?a µ??
  e??a? s??dedeµ??? ? ?a e??a? tet??µµ???, ????? ?a
  ?p???e? ???s?? ?p?s????? t?? V ?? µe t?? ?d?a
  ?d??t?ta (vertex cut set, vertex separating set).
• S??desµ???t?ta ??µß?? VC(G) e??? ??af?µat?? G
  e??a? t? e????st? kV, ?ste t? ???f?µa G ?a
  ??e? ??a s????? ap? k ??µß??? t?µ?? (vertex
  connectivity).
• ??a ???f?µa G ???eta? k-s??dedeµ??? a? VC(G) k.
• k-s??dedeµ??? ? ? d?a??af? k ??lt k ??µß??
  d?µ?????e? µ?-tet??µµ??? s??e?t??? ???f?µa
• k(G) k
• VC(G)

k VC(G)
4
??sa???? (2)

• Te???µa ??a? ??µß?? e??? d??d??? v e??a? ??µß??
• t?µ?? a? ?a? µ???? a? d(v) gt 1.
  ----------------------gt
• ?et??µµ??? ???f?µa
• ?? s??e?t??? ???f?µa
• ????e? G??f?µa

?VC(G)n-1
5
Te???µa
6
??sa???? (3)

• ????sµa ???e ap?? s??dedeµ??? ???f?µa ??e?
  t??????st?? 2 ??µß??? p?? de? e??a? ??µß?? t?µ??.

7
??sa???? (3)

• Te???µa ??a? ??µß?? v e??a? ??µß?? t?µ?? a? ?a?
  µ???? a? ?p?????? 2 ??µß?? u ?a? w (u,wltgtv), ?ste
  ? ??µß?? v ?a ß??s?eta? se ???e µ???p?t? ap? t??
  u p??? t?? w.

8
??sa???? (4)

• S????? a?µ?? t?µ?? ? ? e??a? t? s????? t?? a?µ??
  ?ste t? ???f?µa G-E ?? ?a µ?? e??a? s??dedeµ???,
  ????? ?a ?p???e? ???s?? ?p?s????? t?? ? ? µe t??
  ?d?a ?d??t?ta (edge cut set, edge separating set)
• S??desµ???t?ta a?µ?? EC(G) e??? ??af?µat?? G
  e??a? t? e????st? kE ?, ?ste t? ???f?µa G ?a
  ??e? ??a s????? ap? k a?µ?? t?µ?? (edge
  connectivity)
• ??a ???f?µa G ???eta? k-s??dedeµ??? ?? p??? t??
  a?µ?? a? EC(G)gtk
• Te???µa ??a a?µ? e e??a? a?µ? t?µ?? a? ?a? µ????
  a? ?p?????? 2 ??µß?? u ?a? w, t?t???? ?ste ? a?µ?
  e ?a ß??s?eta? se ???e µ???p?t? ap? t?? ??µß? u
  p??? st?? w.

9
??sa???? (5)

• Te???µa ??a a?µ? e??a? a?µ? t?µ?? a? ?a? µ????
  a? de? pe????eta? se ?????.
• Te???µa Whitney VC(G) lt EC(G) lt d(G).
• ????sµa EC(G) lt floor(2m/n).
• Te???µa ?st? 1lt l lt n-1. ?? d(G) gt (n l
  -2)/2, t?te ? G e??a? l-s??dedeµ????.

10
Te???µa e a?µ? t?µ?? e?? ? de? pe????eta? se
????? - ?p?de???
11
Te???µa e a?µ? t?µ?? e?? ? de? pe????eta? se
?????
u v
P
12
Te???µa e a?µ? t?µ?? e?? ? de? pe????eta? se
?????
13
Te???µa Chartrand Harary, 1968
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?eµ???a G??f?? (1)

• ??a? d?s??dedeµ??? (biconnected) ???f?µa de? ??e?
  ??µß??? t?µ??. ??a t?t??? ???f?µa ap?te?e? ??a
  teµ???? (block) ? µ?a d?s???st?sa (bicomponent).
• ?s?te???? ???a µ???p?t?a (internally disjoint
  paths) e??a? d?? µ???p?t?a µe ??????? te?µat?????
  ??µß???, ????? ???e? ??????? ??µß???.
• Te???µa Whitney ??a ???f?µa e??a? d?s??dedeµ???
  a? ?a? µ???? a? 2 ?p????d?p?te ??µß?? t?? e??a?
  s??dedeµ???? µe t??????st?? 2 e??te???? ???a
  µ???p?t?a.
• ????sµa ?? ??a? ???f?? e??a? d?s??dedeµ????,
  t?te d?? ?p??esd?p?te ????f?? t?? a?????? se ??a?
  ?????

y
u
P1
???e a?µ? se ?????
P2
15
?eµ???a G??f?? (2)

• ????sµa ?? ??a ???f?µa ap?te?e?ta? ap? ??a
  teµ???? µe ngt3, t?te d?? ?p??esd?p?te a?µ?? t??
  a?????? se ??a? ?????.
• Te???µa Menger ? µ???st?? a???µ?? es?te????
  ????? µ???pat??? ap? ??a ??µß? u se ??a ??µß? v
  ?s??ta? µe t?? e????st? a???µ? ??µß?? p??
  ???????? t??? ??µß??? u ?a? v.
• Te???µa ??a ???f?µa e??a? k-s??dedeµ??? a? ?a?
  µ???? a? ??a ta ?e??? ??µß?? e?????ta? µe
  t??????st?? k es?te???? ???a µ???p?t?a.

16
?eµ???a G??f?? (3)

• Te???µa ? µ???st?? a???µ?? es?te???? ?????
  µ???pat??? ap? ??a ??µß? u se ??a ??µß? v ?s??ta?
  µe t?? e????st? a???µ? a?µ?? p?? ???????? t???
  ??µß??? u ?a? v.
• Te???µa ??a ???f?µa e??a? k-s??dedeµ??? ?? p???
  t?? a?µ??, a? ?a? µ???? a? ??a ta ?e??? ??µß??
  e?????ta? µe t??????st?? k es?te???? ???a
  µ???p?t?a.

17
?s?µ??f?sµ?? (1)

• ?e? ?p???e? ap?te?esµat???? a??????µ?? ??a t?
  d?ap?st?s? t?? ?s?µ??f???t?ta? d?? ??af?µ?t??.
• ???t? ??s? (? ?e???te??) ??at??µe t? ??a ???f?µa
  sta?e?? ?a? epa?ad?at?ss??µe t??? ??µß??? t??
  ?????. ??te???µe n2 s?????se??. ??a ?
  p???p????t?ta e??a? t???? ?(n!n2)O(nn).
• ?e?te?? ??s? ?? t? ???f?µa e??a? ap????e?µ??? µe
  p??a?a p??spt?s??, t?te a??e? ? p??a?a? t?? e???
  ??af?µat?? ?a µetat?ape? st?? p??a?a t?? ????? µe
  t? ß???e?a a?t?µeta??se?? ??aµµ?? ?/?a? st????.
• ?p?????? ap?te?esµat???? a??????µ?? ??a e?d????
  pe??pt?se?? ??af?µ?t??.

18
?s?µ??f?sµ?? (2)

• ?e???sµata ??a t?? e????? d?ap?st?s? a? d??
  ??af?µata de? e??a? ?s?µ??f???
• ?d?a t???
• ?d?? µ??e???
• ?d?a a???????a ßa?µ??
• ?d??? a???µ?? s???st?s??
• G?a ???e s???st?sa t?? (4) apa?t??ta? ?et??? ??
  p??te? t?e?? e??t?se??
• ????? ?? d?? ???f?? t? ?d?? ???µat??? p??????µ?
• G?a nlt8, a? ??e? ?? e??t?se?? apa?t????? ?et???,
  t?te ta ??af?µata e??a? ?s?µ??f???.

19
?s?µ??f?sµ?? (3)
y
u
z
y
y
y
z
20
?s?µ??f?sµ?? (4)

• ??? ??af?µata e??a? 1-?s?µ??f??? a? ?a??sta?ta?
  ?s?µ??f??? µet? t?? epa?e????µ??? d??spas? t??
  ??µß?? t?µ??. ?? t? ???f?µa ap?te?e?ta? ap? ??a
  teµ????, t?te ? ?????a t?? 1-?s?µ??f???t?ta?
  ta?t??eta? µe t?? ?????a t?? ?s?µ??f???t?ta?.
• Te???µa ?? d?? ??af?µata e??a? ?s?µ??f???, t?te
  ?s???ta? ? se??? ?a? ? µ?de????t?t? t???.
• ??? ??af?µata ????? a?t?st????a ??????, a?
  ?p???e? aµf?µ???s?µa?t? a?t?st????a a?µ?? ?a?
  ??????.
• Te???µa ??? 1-?s?µ??f??? ??af?µata ?????
  a?t?st????a ??????.

21
?s?µ??f?sµ?? (5)

• ??? ??af?µata e??a? 2-?s?µ??f??? a? ?a??sta?ta?
  ?s?µ??f??? µet? t?? epa?e???µµ??? efa?µ??? t??
  p???e??
• ??asp??ta? t?? a?µ?? t?µ??, ?/?a?
• ?pa?as??d???ta? ta ?p???af?µata a??????ta? ta
  s?µe?a s??des??.
• ??? ?s?µ??f??? ??af?µata e??a? 1-?s?µ??f???,
• d?? 1-?s?µ??f???? ???f?? e??a?
  2-?s?µ??f????.
• ?? a?t??et? de? ?s??e?.

22
?s?µ??f?sµ?? (5)

• Te???µa ??? ???f?? e??a? 2-?s?µ??f???? a? ?a?
  µ???? a? ????? a?t?st????a ??????.

23
???????µ?? (1)

• Bfs-Moore-1959-???s? ?????.
• Dfs-Hopcroft-Tarjan-1973-???s? st??ßa?.
• ???s?µ?p?????ta? ??a ep?s?e?? ??µß??, e??es?
  s???st?s??, e??es? ap?st?se??.

24
???????µ?? (2)

• ?a??de??µa e??es?? ap???pt??s?? ??µß?? (s?µe???
  ?????s??) µe dfs.
• ?pa?te?ta? ? ?p?????sµ?? t?? dfi(v) ?a? l(v) ??a
  ???e v. G?a t?? ??µß? v, t? l(v) e??a? t?
  e????st? t?? t???? p?s?t?t??
• dfi(v)
• l(s), ?p?? t? s ???? t?? v
• dfi(w), ?p?? w e??a? ????f? µe ?p?s??a a?µ? (v,w)

25
???????µ?? (3)

• ??a? ??µß?? v e??a? ??µß?? t?µ?? a?
• ???a? ???a d??d??? ?a? ??e? pe??ss?te?? ap? 1
  pa?d??.
• ?e? e??a? ???a d??d???, a??? ??e? dfi(v) lt l(s),
  ?p?? s ???? t?? v.

26
???????µ?? ??a ??µß??? t?µ??
27
???????µ?? ??a ??µß??? t?µ??
28
Te???µa (??a ??µß??? t?µ??)
29
??µß?? t?µ??
30
Ge???e?µ??? ???ß??µa S??d?sµ?? (1)

• ?? p??ß??µa t?? s??d?sµ?? e??a? t? ???st?
  p??ß??µa t?? e??es?? t?? e????st?? ?e??????t??
  d??d???.
• ??a e????st? ?e?????? d??d?? ??e? s??desµ???t?ta
  ????f??/a?µ?? ?s? µe 1.
• ?? ?e???e?µ??? p??ß??µa t?? s??d?sµ?? e??a? ?a
  ß?e?e? t? ?p????f?µa d????t?? ??af?µat??
  e????st?? ß????, ?ste ? s??desµ???t?ta ?a ?s??ta?
  µe l. ?? l1, t?te ta p??ß??µata ta?t????ta?.

n3
??? a???p?st?
VC(G)1 EC(G)3
VC(H)EC(H)4
31
Ge???e?µ??? ???ß??µa S??d?sµ?? (2)

• Ta ?e????e? ? pe??pt?s? µ? ????sµ???? ???f??.
  S??p?? e??a? ? e??es? t?? p?????? ?p???af?µat??
  H1n µe t?? e????st? a???µ? a?µ?? ?a? ??µß??
  a???µ?µ???? ap? 0 ?? n-1.
• ? a??????µ?? ??e? t?e?? pe??pt?se??
• l ??t?? (l2r).
• l pe??tt? (l2r1), n ??t??.
• l pe??tt? (l2r1), n pe??tt?.

32
Ge???e?µ??? ???ß??µa S??d?sµ?? (3)

• ? a??????µ?? ??e? t?e?? pe??pt?se??
• l ??t?? (l2r). ??? ??µß?? i ?a? j e??a?
  ?e?t??????, a?
• i-r lt j lt
  ir
• l pe??tt? (l2r1), n ??t??. ?atas?e???eta? t?
  ???f?µa H2r,n (efa?µ??eta? ? p??????µe?? s??s?),
  e?? ep?s?? d?? ??µß?? i ?a? in/2 e?????ta? ??a 1
  lt i lt n/2.
• l pe??tt? (l2r1), n pe??tt?. ?atas?e???eta? t?
  ???f?µa H2r,n, e?? ep?s?? e???eta? ? ??µß?? 0 µe
  t??? (n-1)/2 ?a? (n1)/2 ?a? t?? ??µß? l µe t??
  ??µß? i(n1)/2 ??a 1lt i lt (n-1)/2.

33
Ge???e?µ??? ???ß??µa S??d?sµ?? (3)

• Te???µa ?? ???f?µa ?1,n e??a? l-s??dedeµ???.
• Te???µa ? e????st?? a???µ?? a?µ?? t?? ??af?µat??
  ?1,n e??a? ceil(ln/2).

H4,8
H5,8
H5,9