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14 (set) S={v 1, v 2, v 3 }={v 3, v 2, v 1 }={v 1, v 1, v 2, v 3 } (unordered pair) 2 1 {v 1, v 2 } {v 2 } (v 1, v 2 ) (v 2, v 2 ) 14

15 () <> 15

16 G=<V,E> (vertex set) V V(G) (edge set) E E(G) e " eî E( G),( e Î{ 1,2} ) " eî E( G),( e Í V ( G) ) v 1 (order) ν(g)= V(G) // e 1 (size) ε(g)= E(G) v 2 e=(u, v)=uv e 2 e 4 e 5 e 7 v 5 v 3 e 3 v 4 16

17 ( ) v 1 v 2 e 1 (endpoint) v 1 v 2 e 1 (incident) v 1 v 2 (adjacent) e 1 e 2 (adjacent) e 7 (loop) v 1 e 1 v 2 e 5 e 7 e 2 e 4 v 5 v 3 e 3 v 4 17

18 e 1 v 1 v 2 e 5 e 6 e 7 e 2 e 4 v 5 v 3 e 3 v 4 18

19 (multiset) S={v1, v2, v3}={v3, v2, v1} {v1, v1, v2, v3} (multiple edges) e 5 e 6 E(G)={(v 1, v 2 ), (v 2, v 3 ), (v 3, v 4 ), (v 3, v 5 ), (v 1, v 5 ), (v 1, v 5 ), (v 5, v 5 )} e 1 v 1 v 2 e 5 e 6 e 7 e 2 e 4 v 5 v 3 e 3 v 4 19

20 (degree) 2 d(v 1 )=3 d(v 5 )=5 D d ( G) ( G) = max vîv ( G) = min vîv ( G) d d ( v) ( v) e 1 v 1 v 2 e 5 e 6 e 7 e 2 e 4 v 5 v 3 e 3 v 4 20

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22 (null graph) V(G)= (empty graph) (trivial graph) (simple graph) (complete graph) k- (k-regular graph) (bipartite graph) (complete bipartite graph) 22

23 (null graph) (empty graph) E(G)= (trivial graph) (simple graph) (complete graph) k- (k-regular graph) (bipartite graph) (complete bipartite graph) 23

24 (null graph) (empty graph) (trivial graph) ν(g)=1 (simple graph) (complete graph) k- (k-regular graph) (bipartite graph) (complete bipartite graph) 24

25 (null graph) (empty graph) (trivial graph) (simple graph) (complete graph) k- (k-regular graph) (bipartite graph) (complete bipartite graph) v 2 e 2 e 1 v 3 v 1 e 5 e 6 e 7 e 4 v 5 e 3 v 4 25

26 (null graph) (empty graph) (trivial graph) (simple graph) (complete graph) K ν(g) k- (k-regular graph) (bipartite graph) (complete bipartite graph) 26

27 (null graph) (empty graph) (trivial graph) (simple graph) (complete graph) k- (k-regular graph) ( G) ( d( v) k) " v ÎV, = (bipartite graph) (complete bipartite graph) 27

28 (null graph) (empty graph) (trivial graph) (simple graph) (complete graph) k- (k-regular graph) (bipartite graph) ( G) = X ÈY, X ¹ Æ, Y ¹ Æ X ÇY = Æ ÎE( G) (( eç X ¹ Æ) Ù( eç ¹ Æ) ) V, " e, Y (complete bipartite graph) 28

29 (null graph) (empty graph) (trivial graph) (simple graph) (complete graph) k- (k-regular graph) (bipartite graph) (complete bipartite graph) X-Y K X, Y 29

30 (null graph) (empty graph) (trivial graph) (simple graph) (complete graph) k- (k-regular graph) (bipartite graph) (complete bipartite graph) 30

31 31

32 1 H G (subgraph) V(H) V(G) E(H) E(G) H G (spanning subgraph) H G V - (induced subgraph) H G E - (edge-induced subgraph) v 1 e 1 v 2 v 2 e 5 e 2 e 4 v 5 e 2 e 4 v 5 v 3 v 4 v 3 e 3 v 4 32

33 1 H G (subgraph) H G (spanning subgraph) V(H)=V(G) H G V - (induced subgraph) H G E - (edge-induced subgraph) v 1 e 1 v 1 v 2 v 2 e 5 e 2 v 5 e 2 e 4 v 5 v 3 v 4 v 3 e 3 v 4 33

34 1 H G (subgraph) H G (spanning subgraph) H G V - (induced subgraph) ( ) ( ) ( ) ( ) ( )) " v, v ÎV ' = V H, v, v Î E G v, v Î E H H=G[V ] i j i H G E - (edge-induced subgraph) j i j e 1 v 1 v 2 v 2 e 5 e 2 e 4 v 5 e 2 e 4 v 5 v 3 e 3 v 4 v 3 e 3 v 4 34

35 1 H G (subgraph) H G (spanning subgraph) H G V - (induced subgraph) H G E - (edge-induced subgraph) ( ) V H = e H=G[E ] eîe '! = E( H ) v 1 e 1 v 1 e 5 v 2 e 5 e 4 v 5 e 2 e 4 v 5 v 3 v 3 e 3 v 4 35

36 1 ( ) G-V G[V(G)\V ] G-v G-{v} G-E <V(G), E(G)\E > G-e G-{e} v 1 e 1 v 2 v 2 e 5 e 2 e 4 v 5 e 2 e 4 v 5 v 3 e 3 v 4 v 3 e 3 v 4 36

37 2 G H (isomorphism) α: V(G) V(H) (u, v) E(G) iff. (α(u), α(v)) E(H) H v 4 v 2 e 2 e 3 e 4 e1 v e 5 v 1 5 v 2 e 2 e 1 v 1 e 5 e 4 v 5 v 3 v 3 e 3 v 4 37

38 2 ( ) 38

39 10 39

40 2 ( ) NP P NPC 2015 quasi-polynomial time 40

41 G (complement) ( G),{( x y) Ï E( G) } G = V, G H (union) G H / (addition) G H / (join) G H (symmetric difference) b b x a x a c c 41

42 G (complement) G H (union) ( G) ÈV( H ) E( G) È E( H ) G È H = V, G H / (addition) G H / (join) G H (symmetric difference) b b b x a È a y x a y c c c 42

43 G (complement) G H (union) G H / (addition) V G + H = V( G) ÈV( H ), E( G) È E( H ) G H / (join) G H (symmetric difference) ( G) ÇV( H) = Æ b e b e x a + d y x a d y c f c f 43

44 G (complement) G H (union) G H / (addition) G H / (join) V ( G) ÇV( H) = Æ ( G) ÈV( H ), E( G) È E( H ) È{ ( x, y) xîv( G) Ù yîv( H )} G Ú H = V G H (symmetric difference) c a b d Ú x y c a b d x y 44

45 G (complement) G H (union) G H / (addition) G H / (join) G H (symmetric difference) ( ( ) ( )) ( E( G) Ç E( H )) G Å H = V, E G È E H \ ( G) = V( H) V V = b b b c a d Å c a d c a d 45

46 (walk) v 0, e 1, v 1,, e k, v k e i =(v i-1, v i ) v 0 v k e 1 v 1 v 2 e 5 e 2 e 4 v 5 v 3 e 3 v 4 46

47 (walk) v 0, e 1, v 1,, e k, v k e i =(v i-1, v i ) v 0 v k (trail) e 1 v 1 v 2 e 5 e 2 e 4 v 5 v 3 e 3 v 4 47

48 (walk) v 0, e 1, v 1,, e k, v k e i =(v i-1, v i ) v 0 v k (trail) (path) e 1 v 1 v 2 e 5 e 2 e 4 v 5 v 3 e 3 v 4 48

49 (walk) v 0, e 1, v 1,, e k, v k e i =(v i-1, v i ) v 0 v k (trail) (path) (closed walk) v 1 e 1 v 2 e 5 e 2 e 4 v 5 v 3 e 3 v 4 49

50 (walk) v 0, e 1, v 1,, e k, v k e i =(v i-1, v i ) v 0 v k (trail) (path) (closed walk) (closed trail) v 1 e 1 v 2 e 5 e 2 e 4 v 5 v 3 e 3 v 4 50

51 (walk) v 0, e 1, v 1,, e k, v k e i =(v i-1, v i ) v 0 v k (trail) (path) (closed walk) (closed trail) (cycle) e 1 v 1 v 2 e 5 e 2 e 4 v 5 v 3 e 3 v 4 51

52 (length) u v (shortest path) u v (distance) d(u, v) (odd cycle) (even cycle) (girth) (circumference) v 2 e 2 e 1 v 1 e 5 e 4 v 5 v 3 e 3 v 4 52

53 1.1.3 ( ) 3 G d G ³ G 1. G P=v 0,, v k 2. v 0 P v 1 v i v j i j i j v 0 v 1 v i v j v k 53

54 1.1.2 ν X Y X Y u X={ u } Y={ u } v 1 u v 1 v 2 u u' 1 v 2 u u'(v 1 ) v 2 u 54

55 (connected) (connected graph) (connected component) v 1 v 3 v 4 G w(g) v 2 v 5 55

56 1.1.2 ( ) ν X Y X Y u X={ u } Y={ u } u u' v 1 1 u v 1 v 2 u v 2 56

57 G ε(g) ν(g)-1 1. w 1 e 1 v 1 v 2 e 5 e 2 e 4 v 5 v 3 e 3 v 4 57

58 (eccentricity) (center) (radius) (diameter) (median) e ( v) rad = max uîv arg mine vîv ( G ) ( G) ( ) diam G arg min vîv ( G) ( G) ( v) vîv d = min ( v, u) ( G) e = max e å uîv vîv ( G) ( G) ( v) ( v) ( v u) d, v 1 v 6 v 2 v 3 v 4 v 5 v 7 v 10 v 13 v 9 v 8 v 11 v 14 v 12 58

59 1.4 // 1.35 ( ) // 1.23 // 1.31 // 1.63 // 62