graph theory chapter 8 hamiltonian graphs 大葉大學 (da-yeh univ.) 資訊工程系 (dept. csie)...

Graph TheoryGraph TheoryChapter 8Chapter 8 Hamiltonian GraphsHamiltonian Graphs

大葉大學大葉大學 (Da-Yeh (Da-Yeh Univ.)Univ.)資訊工程系資訊工程系 (Dept. (Dept. CSIE)CSIE)黃鈴玲黃鈴玲 (Lingling (Lingling Huang)Huang)

Copyright Copyright 黃鈴玲黃鈴玲Ch8-Ch8-22

OutlineOutline

8.1 An Introduction to 8.1 An Introduction to Hamiltonian GraphsHamiltonian Graphs

8.28.2 Which Graphs are Which Graphs are Hamiltonian?Hamiltonian?

8.3 The Traveling Salesman 8.3 The Traveling Salesman ProblemProblem


8.1 An Introduction to 8.1 An Introduction to Hamiltonian GraphsHamiltonian GraphsDefinition: A graph of G is called Hamiltonian if G

has a spanning cycle.

A dodecahedron ( 正 12 面體 )

A hamiltonian cycle


KnightKnight’’s Tour Puzzles Tour Puzzle

Is it possible for a knight to tour the chessboard (nn), that is, visit each squareexactly once and return to its initial square?

4 4 chessboard: 此走法不存在8 8 chessboard: 存在 , see Fig 8-2


pf: (By induction on n) Base: (n=3) OK Inductive: Assume the (n1)-cube is

hamiltonian. Consider the n-cube:

Ex1: Show that the graph of the n-cube is hamiltonian.

3 - cube

(n 1)-cube (n 1)-cube

xy y’

x’


Ex2Ex2: Show that the graph : Show that the graph GG below is below is not hamiltonian. not hamiltonian.

hamiltonian cycle hamiltonian cycle 必定包含這些必定包含這些edge.edge.

Hint:Hint: 考慮通過此三點的考慮通過此三點的edgeedge

本身已構成本身已構成 cyclecycle ，，故不可能故不可能 hamiltonian hamiltonian


Observe:1. Cn (n3) is hamiltonian.hamiltonian.2. Kn (n3) is hamiltonian.hamiltonian.3. A connected graph with cut-vertices is not hamiltonian.hamiltonian.


G1

G3

Gn

G2

pf:pf: G G is hamiltonian is hamiltonian a hamiltonian cycle a hamiltonian cycle CC

Suppose Suppose kk((GGSS) = ) = nn, where , where GG11, , GG22,,……, , GGnn are components of are components of GGS.S.

Let Let uuii (1 (1 ii nn) be the last vertex of ) be the last vertex of CC that belongs to that belongs to GGii..

Let Let vvii be the vertex be the vertex that immediately that immediately follows follows uuii on on CC..((uuii 的下一點即是的下一點即是 vvii 且且 vvii SS))

Thm 8.1:Thm 8.1: If If GG is hamiltonian, than is hamiltonian, than kk((G G SS) ) | |SS| | for every for every SS VV((GG), ), SS ..

C…

v1

v3

vn

v2

u1

un

u3

u2


(1) (1) vvii SS, , ii (2) (2) vvjj vvk k ,, if if jj kk ||SS| | nn ∴ ∴ kk((GGSS) ) | |S|.|.

G1

G3

Gn

G2

C

…

v1

v3

vn

v2

u1

un

u3

u2


pf: pf: 例如例如 KK2,32,3

上與下兩個上與下兩個 part part 點數不一致點數不一致∴∴無法有無法有 hamiltonian cyclehamiltonian cycle

Ex5:Ex5: Show that no bipartite graph Show that no bipartite graph of odd order is hamiltonian.of odd order is hamiltonian.


HomeworkHomework

Exercise 8.1:Exercise 8.1: 1, 2, 3, 4, 5 1, 2, 3, 4, 5


OutlineOutline

8.1 An Introduction to 8.1 An Introduction to Hamiltonian Graphs Hamiltonian Graphs





Thm 8.2 (Dirac’s Thm)G: order p 3If deg(v) p/2 v V(G), then G is Hamiltonian.

Pf: (1) If p = 3 degdeg((vv) ) 22 vv VV((GG) ) GG CC33

(2) If p 44, , let let PP: : vv1 1 ,v,v2 2 ,…, ,…, vvn n be a longest path in G.

v1 v2 v3 vn-1

vn


Then every neighbor of v1 and vn is on P.

∵ ∵ deg(v) p/2 ∴ ∴ nn 1+p/2

Claim: vi V(P), 2 i n, s. t., v1~ vi and vi1~ vn .

pf of claim:

If not, vj with vj ~ v1 vj1 ~ vn

∵ ∵ deg(v1) p/2

∴ ∴v1 到 vn1 中至少有至少有 pp/2/2 個點與個點與 vn 不相連不相連 ∵ ∵ deg(vn) p/2

∴ ∴ 此此 pathpath 至少有至少有 p/2 ++ p/2 + + 11 個點個點

v1

v2 v3vi1

vnvn-1vi

(vn)


Let C: v1, v2, …, vi1, vn, vn1, …, vi, v1

C is a cycle of length n.If n<p then u V(GC) ∵ ∵ deg(u) p/2, |V(C)| = n 1+ p/2∴ ∴ vk s.t. u ~ vk (1 k n)∴ ∴ CC∪{∪{uvk}} contains a path longer than

P. n n = = pp C is a Hamiltonian cycle.※ Thm8.2 中 deg(v) p/2 的條件若改為 (p1)/2 則不保證存在 hamiltonian cycle. 例 : Kn, n+1

v1

v2 v3vi1

vnvn-1vi


※Thm8.2 不能 cover 的例子 : (1) Cn (n 5): hamiltonian p 5, deg(vi) = 2< p/2 i

(2) +(Km U Kn): hamiltonian2K

Fig 8-5: +(K2 U K3) 2K

p = m + n + 2Suppose m < n,deg(v) = m + 1 < p/2 if vV(Km)


Thm8.3 Let G be a graph of order p 3.

Suppose u and v are nonadjacent vertices of G s.t. deg(u) + deg(v) p. Then G is hamiltonian iff G+uv is hamiltonian.

Pf: “”: trivial. “”: Let C be a hamiltonian cycle of G+uv. If uv C G is hamiltonian. If uv C spanning u-v path P (P = C uv)


P :

(P 上使用了所有點 ) Similar as the proof of Thm8.2, G has a

hamiltonian cycle.

u v


上述定理提供了判斷圖形是否 hamiltonian 的一種方法重複將不相連但 degree 和 p 的兩點連一條邊新圖形是否 hamiltonian 決定了原圖是否 hamiltonian

Fig 8-6

p=7 Not hamiltonian!


Def:Def: A A closureclosure cc((GG)) of a graph of a graph GG of of

order order pp is a graph obtained from is a graph obtained from GG by recursively joining pairs of by recursively joining pairs of nonadjacent vertices whose nonadjacent vertices whose degree sum is at least degree sum is at least pp until no until no such pairs remain.such pairs remain.

※ cc((GG)) is unique.is unique.


Thm8.5Thm8.5 GG is hamiltonianis hamiltonian cc((GG)) is hamiltonian.is hamiltonian.

Cor8.5(a) Let G be a graph of order p3. If c(G) KKp p , , then then GG is hamiltonian. is hamiltonian.Fig 8.7

G is hamiltonian.c(G) KK99G


Cor8.5(b) |V(G)| = p 3. u,v V(G). If deg(u) + deg(v) p u ~ v,

then then GG is hamiltonian.is hamiltonian.Cor8.5(c) |V(G)| = p 3. Let d1, d2, …dp be the degree sequence

of G with d1 d2 … dp. Suppose for every k < p/2,dk k dpk p k. ThenThen GG is hamiltonian.is hamiltonian.

Cor8.5(d) |V(G)| = p, |E(G)| = q. If q , thenthen GG is is

hamiltonian.hamiltonian.2

21

p


Ex5.Ex5. Show that if a graph of order Show that if a graph of order at least 3 has an isolated vertex or an end-3 has an isolated vertex or an end-vertex, then its closure is not complete.vertex, then its closure is not complete.

pf:(1) If u is an isolated vertex of graph G, then deg(u) = 0, deg(v) p2 for every vertex v u.

In c(G), u ~ v u v c(G) Kp

(2) If u is an end-vertex then deg(u) = 1, deg(v) p2 if u ~ v c(G) Kp


HomeworkHomework

Exercise 8.2:Exercise 8.2: 1, 3, 4, 8, 9, 11 1, 3, 4, 8, 9, 11


OutlineOutline

8.1 An Introduction to 8.1 An Introduction to Hamiltonian Graphs Hamiltonian Graphs




8.3 The Traveling 8.3 The Traveling Salesman ProblemSalesman Problem

GG:: connected weighted graph,connected weighted graph, vvi i VV((GG)): the cities,: the cities, ww((vviivvjj)) of edgeof edge vviivvjj:: the distance to travel the distance to travel directly between directly between vvii andand vvj j .. (Assume that(Assume that GG is complete)is complete)

TSP: Suppose that a salesman is required to make a round trip through a given collection of p(3) cities. What route should he take to minimize the total distance traveled?

※ ※ TSP asks for a Hamiltonian cycle of minimum weight.TSP asks for a Hamiltonian cycle of minimum weight.


此處提供兩種作法此處提供兩種作法前提前提 : : 需先符合需先符合 triangle inequality (triangle inequality ( 三角不等三角不等

式式 )) ww((vvii,v,vkk) ) ww((vvii,v,vjj) + ) + ww((vvjj,v,vkk))

vi

vj

vk

∵∵NP-complete NP-complete ∴∴改成改成 find low weight find low weight 的的 HCHCTSP:TSP: Given a weighted complete graph Given a weighted complete graph G G and a positive constant and a positive constant BB, does there , does there exist a hamiltonian cycle exist a hamiltonian cycle CC in in GG so that so that ww((CC) ) BB??


Algorithm 8.1 Algorithm 8.1 (a greedy algorithm)(a greedy algorithm)[ To determine a low weight HC in a weighted complete graph G of order p3 satisfying the triangle inequality. ]1. n 1. (n is the cycle length) 2. Select any vertex of G to form Cn.

(Cn 剛開始只有一個點 )3. If n < p, then find a vertex vn not on Cn s.t.

w(unvn) is minimum for some un is on Cn, and go to Step 4. Otherwise, Cn is the desired HC.

4. Let Cn+1 be the (n+1)-cycle obtained by inserting vn immediately before un on Cn.

5. n n +1 and return to Step 3.


1. 1. CC11: : vv11

2.2. ∵ ∵ vv22,, ……,, vv66 中，中， ww((vv11vv44)) 最小最小 ∴∴CC22: : vv11vv44vv11

3.3.

∵ ∵ ww((vv33vv44) ) 最小最小 ∴∴CC33: : vv11vv33vv44vv1 1

(( 加在要連的點之前加在要連的點之前 ))

0 4 5 4 5 34 0 5 4 5 75 5 0 1 4 24 4 1 0 3 35 5 4 3 0 33 7 2 3 3 0

M

v1 v2 v3 v4 v5 v6v1

v2 v3 v4 v5 v6

Fig 8-10Fig 8-10

v2 v3 v5 v6

v1

v4 55143733


v2 v5 v6

v1

v3

v4

3 7 3

3 4 4 4 5 5

7 3

5 5 4 4 5 5

v5 v6

v1

v2

v3

v4

v5 v1

v2

v3

v4

v6

7 5 4 5 4

∵ ∵ ww((vv11vv22)) 最小最小∴∴CC44: : vv11vv33vv44vv22vv11

5.5. 6.6.

CC55:: vv11vv33vv44vv22vv66vv11 CC66: : vv11vv55vv33vv44vv22vv66vv11

4.4.

CC66 的的 weight weight 總和為總和為 2424, , 而而 min weight min weight 為為 1818..若改選別的點當若改選別的點當 CC11,, 可能可能 weight weight 總和總和更小更小 ..


Thm 8.8Thm 8.8 C :: a a HC given by Algorithm 8.1given by Algorithm 8.1 Cm :: min weightmin weight HC w((C)) 2 2 w((Cm))

(Algorithm 8.1(Algorithm 8.1 不保證能找出不保證能找出 min HCmin HC,, 但用但用 Algorithm 8.1 Algorithm 8.1 找出的找出的 cycle cycle 其其 weightweight 不會大於不會大於 min HC min HC 的兩倍的兩倍 .).)


Algorithm8.2 Algorithm8.2 (( 利用利用 min spanning tree)min spanning tree)

1. Find a min spanning tree1. Find a min spanning tree TT of of G.G.2. Conduct a depth-first search of2. Conduct a depth-first search of T. T.

( ( 起點為起點為 T T 的的 leaf)leaf)3. If3. If vvii11

, , vvii22, …, , …, vviipp

is the order in which the is the order in which the vertices of vertices of TT are visited in step 2, thenare visited in step 2, then output the hamiltonian cycleoutput the hamiltonian cycle vvii11

, , vvii22, …, , …, vviipp

, , vvii11..

[ To determine a low weight HC in a weighted complete graph G of order p3 satisfying the triangle inequality. ]

(Algorithm 8.2 (Algorithm 8.2 找出的找出的 cycle cycle 其其 weight weight

也不會大於也不會大於 min HC min HC 的兩倍的兩倍 .).)


Fig 8-13 ( 原圖同 Fig 8-10)

C: v2,v1,v4,v3,v5,v6,v2

v2

v5

v4

v3

v1

v6

A min spanning tree T(a)

3 3

42

15

3

4

2

6

A depth-first search從 v2 開始

(b)

1 v2

v5

v4

v3

v1

v6

A HC(c)

53

42

1

4

weight weight 總和為總和為 1919


Exercise 8Exercise 8Use Alg. 8.1 and 8.2 to find a closed walk Use Alg. 8.1 and 8.2 to find a closed walk

whose weight does not exceed twice whose weight does not exceed twice the weight of a shortest closed walk in the weight of a shortest closed walk in the given weighted graph the given weighted graph GG..

Sol:Sol: 先把先把 G G 變成變成 completecomplete

Gv2v5

v4 v3

v1

4

1

3

23

5

4

5 73

Gv2v5

v4 v3

v1

1

3

23

5

4

0433140575350323730415240

M

v1 v2 v3 v4 v5v1

v2 v3 v4 v5


HomeworkHomework

Exercise 8.3:Exercise 8.3: 1, 3, 4, 8 1, 3, 4, 8

graph theory chapter 8 hamiltonian graphs 大葉大學 (da-yeh univ.) 資訊工程系 (dept. csie)...

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