# Paths. Connectivity. Euler and Hamilton Paths. Planar graphs.

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1 Pths.. Eulr n Hmilton Pths..

2 Pth D. A pth rom s to t is squn o gs {x 0, x 1 }, {x 1, x 2 },... {x n 1, x n }, whr x 0 = s, n x n = t. D. Th lngth o pth is th numr o gs in it. {, } {, } {, } {, } {, } {, } {, } p. 2

3 Simpl pth. Cyl D. A simpl pth is pth tht os not ontin th sm g mor thn on. D. A pth is ll yl (or iruit) i its irst n lst vrtis r th sm, n its lngth is grtr thn 0. D. A simpl syl is yl tht os not ontin th sm g mor thn on. p. 3

4 Pths n yls in irt grphs? Thr r similr initions or pths n yls in irt grphs. p. 4

5 Connt grph D. An unirt grph is ll onnt i thr is pth twn vry pir o istint vrtis o th grph. p. 5

6 Connt ompomnts g h j i D. A onnt omponnt o grph G is onnt sugrph o G tht is not propr sugrph o nothr onnt sugrph o G. (So, onnt omponnt is mximl onnt sugrph) Qustion: How mny onnt omponnts is in th grph? p. 6

7 Vrtx ut g h D. A vrtx ut V is sust o vrtis, suh tht th grph oms isonnt, i V n thir inint gs r rmov. p. 7

8 Vrtx ut g g h h D. A vrtx ut V is sust o vrtis, suh tht th grph oms isonnt, i V n thir inint gs r rmov. Exmpl: V = {}. This is on o thr minimum vrtx uts in this grph. Cn you in th othr two? p. 8

9 Vrtx ut D. A vrtx ut V is sust o vrtis, suh tht th grph oms isonnt, i V n thir inint gs r rmov. Fin vrtx ut. p. 9

10 Eg ut g h D. An g ut E is sust o gs, suh tht th grph oms isonnt, i th gs E r rmov. p. 10

11 Eg ut g h D. An g ut E is sust o gs, suh tht th grph oms isonnt, i th gs E r rmov. p. 11

12 Distn n imtr g h D. Th istn twn two vrtis in grph is th lngth o th shortst pth twn thm. istn(, g) = 2 D. Th imtr o grph is th istn twn th two vrtis tht r rthst prt. imtr = 3 p. 12

13 Eulr pth n yl D. An Eulr yl in grph G is simpl yl ontining vry g o G. Similrly, n Eulr pth in G is simpl pth ontining vry g o G. (In simpl pth (or yl), gs r not rpt) p. 13

14 Eulr yl Wlk ross ll th rigs on. An gt k to th originl lotion. Wht i w uil two nw rigs? p. 14

15 Eulr yl Wlk ross ll th rigs on. An gt k to th originl lotion. Wht i w uil two nw rigs? p. 15

16 Osrvtion Lt s sy tht w ross rig to th vrtx. Wht is th onition to ontinu wlking? Thr shoul t lst on mor rig t th vrtx. p. 16

17 Osrvtion Lt s sy tht w ross rig to th vrtx. Wht is th onition to ontinu wlking? Thr shoul t lst on mor rig t th vrtx. p. 17

18 Osrvtion Whn w ntr vrtx n thn lv it, w us two rigs. So, vry tim w visit vrtx, two rigs r gon. p. 18

19 Fining n Eulr yl I w visit vrtx, w us two rigs. I thr is n vn numr o rigs t th vrtx, thn tr our visit, thr is still n vn numr o rigs. I vrtx hs only on rig, it n only th inl point in th pth. p. 19

20 Nssry n suiint onition or Eulr yls Thorm. A onnt multigrph with t lst two vrtis hs n Eulr yl i n only i h o its vrtis hs vn gr. p. 20

21 Nssry n suiint onition or Eulr yls Thorm. A onnt multigrph with t lst two vrtis hs n Eulr yl i n only i h o its vrtis hs vn gr. j i h g Construting n Eulrin yl tks linr tim in th numr o gs! This is iint. k p. 21

22 Eulr pth Thorm. A onnt multigrph hs n Eulr pth ut not n Eulr yl i n only i it hs xtly two vrtis o o gr. p. 22

23 Iosin Puzzl A puzzl invnt in 1857 y Sir Willim Rown Hmilton: Th tsk is to trvl long th gs o ohron, visit h o 20 vrtis xtly on, n n k t th irst vrtx. p. 23

24 Iosin Puzzl A puzzl invnt in 1857 y Sir Willim Rown Hmilton: Th tsk is to trvl long th gs o ohron, visit h o 20 vrtis xtly on, n n k t th irst vrtx. p. 24

25 Iosin Puzzl A puzzl invnt in 1857 y Sir Willim Rown Hmilton: Th tsk is to trvl long th gs o ohron, visit h o 20 vrtis xtly on, n n k t th irst vrtx. p. 25

26 Hmilton pth g D. A simpl pth in grph G tht psss through vry vrtx xtly on is ll Hmilton pth. An simpl yl in grph G tht psss through vry vrtx xtly on is ll Hmilton yl. p. 26

27 Suiint onitions or yl Thorm (Dir s thorm). I G is simpl grph with n vrtis with n 3 suh tht th gr o vry vrtx in G is t lst n/2, thn G hs Hmilton yl. Thorm (Or s thorm). I G is simpl grph with n vrtis with n 3 suh tht g(u) + g(v) n or vry pir o nonjnt vrtis u n v in G, thn G hs Hmilton yl. p. 27

28 Algorithm or ining yl? Th st lgorithms known or ining Hmilton yl in grph or trmining tht no suh yl xists hv xponntil worst-s tim omplxity in th numr o vrtis o th grph. In t, this is n NP-omplt prolm. p. 28

29 Mor Hmilton yls Th mous Trvling Slsprson Prolm (TSP): Fin th shortst rout trvling slsprson shoul tk to visit givn st o itis. It rus to ining Hmilton yl on omplt grph suh tht th totl wight o th pth is th smllst. p. 29

30 Qustion: Is it possil to join ths houss n utilitis so tht non o th onntions ross? Hous 1 Hous 2 Hous 3 Gs Wtr Eltriity This is omplt iprtit grph, not y K 3,3. p. 30

31 Qustion: Is it possil to join ths houss n utilitis so tht non o th onntions ross? Hous 1 Hous 2 Hous 3 Gs Wtr Eltriity This is omplt iprtit grph, not y K 3,3. p. 31

32 D. A grph is ll plnr i it n rwn in th pln without ny gs rossing. Complt grph K 4 is plnr: p. 32

33 D. A grph is ll plnr i it n rwn in th pln without ny gs rossing. 3-imnsionl hypru grph, Q 3, is plnr: h g h g p. 33

34 Eulr ormul A rwing o plnr grph ivis th pln into s, rgions oun y gs o th grph. our s: Thorm (Eulr ormul). Lt G onnt plnr simpl grph with gs n v vrtis. Lt r th numr o s in plnr rprsnttion o G. Thn v + = 2. p. 34

35 Thorm (Kurtowski). A grph is plnr i n only i it os not ontin suivision o K 3,3 or K 5. Wht is suivision? Insrting nw vrtx into n xisting g o grph is ll suiviing th g, n on or mor suivisions o gs rt suivision o th originl grph. K 3,3 K 5 p. 35

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