Tea Gaft Teesefa

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

• ?ef??a?? 1 ??sa????

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?st?????

• 1736 Euler, ??f??e? Koenigsburg
• 1847 Kirchoff, d??d?a, ??e?t???? d??t?a
• 1847 Cayley, d??d?a, ?s?µe??
• ?d?????a??????? CnH2n2
• 1850 Cayley - De Morgan - Moebius,
• ???µat?sµ?? µe 4 ???µata
• 1859 Hamilton, d?de??ed??
• 1936 Koenig, t? p??t? ß?ß???

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?fa?µ???? Te???a G?af?µ?t??
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???????µ?? Te???a G?af?µ?t??
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?as???? ?????e? (1)

• ???sµ?? s????? ??µß?? (????f??) ?a? a?µ??
• S?µß???sµ?? G(V, E), G(V , E), (V(G) , E(G)),
  (n,m), (p,q)
• V n t???-order
• E m µ??e???-size
• ?epe?asµ??? ???f?µa n, m pepe?asµ??a
• ?pe??? ???f?µa
• ??d???? pe??pt?se?? n 0 ?e??-empty
• n 1 tet??µµ???-trivial
• m 0 µ?de????-null (Nn)

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?as???? ?????e? (2)

• ?e?µat???? ??µß?? ?a? p??sp?pt??sa a?µ?
• Ge?t?????? ??µß?? a?e???t?t?? ??µß??
• ?????t? s????? ?e?t??as?? ??µß?? ?(v)
• ??e?st? s????? ?e?t??as?? ??µß?? ?v
• ?a?µ?? ??µß?? v degree d(v)
• ?s??e? d(v) N(v)
• ?????st?? ?a? µ???st?? ßa?µ?? ??af?µat?? d(G),
  D(G)

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?as???? ?????e? (3)

• ?a?t??? ??af?µata (regular)
• ??????? ???f?µa (Cn) ???? ?? ??µß?? d(v)2
  (??ß???? k3)
• ??at????? ??af?µata (tet??ed??, ??ß??, ??t?ed??,
  d?de??ed??, e???s?ed??)

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?as???? ?????e? (4)

• ?p?µ???µ???? ??µß?? isolated d(v)0
• ????eµ?? ??µß?? pendant d(v)1
• S??dedeµ??e? s???st?se? (k)
• S??dedeµ??? ???f?µa
• S??dedeµ???? ???f?µa ?at? e????st? t??p?
• Se??? rank (r n-k)
• ??de????t?ta nullity (µ m-n-k)
• ??????, pa??????e? a?µ??
• ?p?? ???f?µa, ?e?d????f?µa, p??????f?µa,
  ?p??e?µe??, ?ate?????µe??/p??sa?at???sµ??? D(V,
  A) ?a? t??a

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?as???? ?????e? (4)
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?as???? ?????e? (5)

• ??µµa t?? ?e??a???? (Sd(v) 2E)
• ????sµa ??a ta?t??? ??af?µata (kv 2E)
• Te???µa ?? p????? t?? pe??tt?? ??µß?? e??a?
  ??t??
• G?aµµ??? ???f?µa L(G)
• ?????? ???f?µa ?n
• ????a ??af?µat??, a???µ?? ????a? (t???)

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?as???? ?????e? (5)
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?as???? ?????e? (5)

• Te???µa ??a p???e? ???f?µa ??e? n(n-1)/2 a?µ??
• Te???µa n-k lt m lt (n-k)(n-k1)/2
• ????sµa ???e ap?? ???f?µa µe (n-1)(n-2)/2 a?µ??
  e??a? s??dedeµ???

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?as???? ?????e? (6)

• ????sµ??? ???f?µa, ß???? a?µ??
• ?s?µ??f??? ??af?µata, ?sa ??af?µata
  ep???af?/et???ta
• Te???µa ?? p????? t?? d?af??et???? ??af?µ?t?? µe
  n ????f?? ?a? m a?µ?? e??a? comb(n(n-1)/2,m)
• ????sµa ?? p????? t?? d?af??et???? ??af?µ?t?? µe
  n ????f?? e??a? 2n(n-1)/2
• ?p????f?µa, ?pe????f?µa, ?e?????? ?p????f?µa,
  ep??easµ??? ap? s????? ????f??/a?µ??

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?as???? ?????e? (6)
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?as???? ?????e? (6)
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????e?? (1)

• ???s? union (G1 U G2)
• ??µ? intersection (G1 n G2)
• ?????sµa da?t????? ring sum
• S?µp????µa complement
• ??a??af? ??µß??/a?µ?? deletion (G-e, G-v)
• ??ta??a?? a?µ?? interchange
• ???spas? ??µß?? split
• ??µß?? t?µ??, s?µe?? ?????s?? articulation
  point
• ?p???pt??sa a?µ?, ??f??a - bridge, ?s?µ??
• ??s??af?? bicoherent, d?as??dedeµ???
  biconnected

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????e?? U n
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?????sµa G1G2
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????e?? (2)

• S??des?, ?????sµa join, sum
• ?????e?d?? ???f?µa wheel graph (?1 Cn-1 Wn)
• ??µe???, d????f?µa, p???µe??? ???f?µa, p???e?
  d?µe???, p???e? p???µe??? (aste??e?d?? ???f?µa
  ?1,n)
• ?a?tes?a?? ????µe?? G1 x G2
• ?e??????af??? ????µe?? G1G2
• S?????e?s? ??µß?? fusion/merge
• ?p?d?a??es? a?µ?? subdivision

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?a?tes?a?? G???µe?? G1 x G2
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?e??????af??? G???µe?? G1 G2
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???????µ?? ?a? G?af?µata

• ???????µ??? ?e???a ??af?µ?t??
• ????p????t?ta ????? ?a? ??????
• S?µß???sµ?? ?
• ?????s? µ?s?? ?a? ?e???te??? pe??pt?s??

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?p????e?s? G?af?µ?t??

• Stat???? a?apa?ast?se??
• ???a?a? ????f?? adjacency matrix
• ???a?a? p??spt?se?? incidence matrix
• ???aµ???? a?apa?ast?se??
• ??ste? a?µ?? edge lists (??a a?a???? ???f???)
• ??ste? ?e?t??as??

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????????a ?a?µ?? (1)

• ?? a????sa a???????a a?e?a??? S
• G?af??? a???????a (S gt G)
• Ea? S gt G, t?te G ???µ??eta? p?a?µat?p???s?
  (realization) t?? a???????a S
• S?????e?
• ?? a???t???? t?µ??
• ?????? pe??tt?? ßa?µ?? ??t??
• ??µ?? µ????te?e? ap? n
• ?a??de??µa 2, 2, 2, 2, 2, 2

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????????a ?a?µ?? (2)

• ???t?s? e??a? ?? s?????e? a?t?? ??a??? ??a ?a
  a?t?st?????eta? µ?a a???????a ßa?µ?? S se
  µ??ad??? ???f?µa G?
• ?a??de??µa 2, 2, 2, 2, 2, 2
• ?p?? (simple) ??af??? a???????a
• e?? e??a? ??af??? ?a? ?p???e? µ??ad???
  p?a?µat?p???s? t??

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????????a ?a?µ?? (3)

• Te???µa ??a a???????a ßa?µ??
• d1, d2, , dn
• e??a? ??af???, e?? ?a? µ??? e?? ? a???????a
• d2-1, d3-1, , dd11-1, dd12, , dn
• e??a? ??af???.

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Te???µa
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Te???µa
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Te???µa
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????????a ?a?µ?? (4)

• ???????µ??
• ?? ??p??? d gt n-1 ? OXI
• ?? ??a e??a? µ?de???? ? OXI
• ?? ??p??? e??a? a???t??? ? OXI
• ??ad??ta?? t?? a???????a? ?at? µ? a????sa t???
• ??a???feta? ? p??t?? ???? (d1) ?a? afa??e?ta? µ?a
  µ???da ap? t??? ep?µe???? d1 ?????
• ???a??e st? ??µa 2
• ?a??de??µa 5, 4, 4, 3, 2, 1, 1

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????????a ?a?µ?? (5)

• ?a??de??µa 5, 4, 4, 3, 2, 1, 1
• S0 5, 4, 4, 3, 2, 1, 1
• B?µa 1 S1 3, 3, 2, 1, 0, 1 gt 3, 3, 2, 1,
  1, 0
• ??µa 2 S2 2, 1, 0, 0, 1 gt 2, 1, 1, 0, 0
• ??µa 3 S3 0, 0, 0, 0

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????????a ?a?µ?? (5)
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????????a ?a?µ?? (6)

• Te???µa ??a a???????a ßa?µ??
• d1, d2, , dn
• e??a? ??af???, e?? ?a? µ??? e??
• (i) S di e??a? ??t??, ?a?
• (ii) S1 lt i lt k di lt k(k-1) S1 lt k1
  lt n min(k, di)
• ??a ???e 1lt k lt n-1.

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????????a ?a?µ?? (7)
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?as???? ?????e? (Perfect)
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????????? ?etaßat??? ?d??t?ta
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????????? ?a? Comparability G?af?µata
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Circular arc G?af?µata
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?ntersection G?af?µata
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Permutation G?af?µata
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???e?a G?af?µata
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???e?a G?af?µata
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???e?a G?af?µata
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???e?a G?af?µata
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???e?a G?af?µata
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???e?a G?af?µata
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???e?a G?af?µata
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Vertex Separators
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Te???µa
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Te???µa s????e?a
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