I118 Graphs and Automata

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1 I8 Graphs and Automata Takako Nemoto t-nemoto/teaching/203--.html April 23

2 0. Û. Û ÒÈÓ 2. Ø ÈÌ (a) ÏÛ Í (b) Ø Ó Ë (c) ÒÑ ÈÌ (d) ÒÌ (e) É Ö ÈÌ 3. ÈÌ (a) Î ÎÖ Í (b) ÒÌ

3 . Û Ñ ÐÒ f : A B, C i A, C 0 C C 0,C 0 C C ËÚÖØ. f(c 0 C ) f(c 0 ) f(c ) ËÚÖØ.

4 ÓÚ ÏÈ 3 ÑÌË ÈÌ ÍÛ, ÛÛ ÓÚ ËÉ ÈÌ Í ÛÙ R. J. È ËÉ ØÛÊÐÖÎÈÊÐØ ÈÔ ÖÐ Ï Ó 7 ÛÍÒ ÚÈÉÊ Ë.

5 2. Ø ÈÌ ÊÏ Ø ÈÌ (undirected graph) (V,E) Ó, V Ó E [V] 2 ÌËÈÍË. ÕÕ [V] 2 = {{x,y} : x V y V}.

6 2. Ø ÈÌ ÊÏ Ø ÈÌ (undirected graph) (V,E) Ó, V Ó E [V] 2 ÌËÈÍË. ÕÕ [V] 2 = {{x,y} : x V y V}. Ú V = {,2,3,},E = {{,2},{2,3},{3,},{,3},{2,},{,}} 2 3 Ú V = {,2,3,},E = {{},{2,3}}

7 2a. Î ÎÖ Í Ø ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node)

8 2a. Î ÎÖ Í Ø ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) E ÛÎ Ì (edge)

9 2a. Î ÎÖ Í Ø ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) E ÛÎ Ì (edge) {a,b} E ÛÓ a Ó b Ñ (adjacent) Ú Ë a Ó b Ì ab(= {a,b}) Õ (incident) Ú Ë

10 2a. Î ÎÖ Í Ø ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) E ÛÎ Ì (edge) {a,b} E ÛÓ a Ó b Ñ (adjacent) Ú Ë a Ó b Ì ab(= {a,b}) Õ (incident) Ú Ë Í a Ú aa ËÈÎ (loop)

11 2a. Î ÎÖ Í Ø ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) E ÛÎ Ì (edge) {a,b} E ÛÓ a Ó b Ñ (adjacent) Ú Ë a Ó b Ì ab(= {a,b}) Õ (incident) Ú Ë Í a Ú aa ËÈÎ (loop) Í v Ñ ÕÚ ËÌÛÒ ÛÒ (degree)

12 2a. Î ÎÖ Í Ø ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) E ÛÎ Ì (edge) {a,b} E ÛÓ a Ó b Ñ (adjacent) Ú Ë a Ó b Ì ab(= {a,b}) Õ (incident) Ú Ë Í a Ú aa ËÈÎ (loop) Í v Ñ ÕÚ ËÌÛÒ ÛÒ (degree) ÛÒ 0 ÛÍ ÔÌÍ (isolated point)

13 2a. Î ÎÖ Í Ø ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) E ÛÎ Ì (edge) {a,b} E ÛÓ a Ó b Ñ (adjacent) Ú Ë a Ó b Ì ab(= {a,b}) Õ (incident) Ú Ë Í a Ú aa ËÈÎ (loop) Í v Ñ ÕÚ ËÌÛÒ ÛÒ (degree) ÛÒ 0 ÛÍ ÔÌÍ (isolated point) ÛÒ ÛÍ ÈÍ (end-vertix) Ó È.

14 2a. Î ÎÖ Í Ø ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) E ÛÎ Ì (edge) {a,b} E ÛÓ a Ó b Ñ (adjacent) Ú Ë a Ó b Ì ab(= {a,b}) Õ (incident) Ú Ë Í a Ú aa ËÈÎ (loop) Í v Ñ ÕÚ ËÌÛÒ ÛÒ (degree) ÛÒ 0 ÛÍ ÔÌÍ (isolated point) ÛÒ ÛÍ ÈÍ (end-vertix) Ó È. ÎÌ a Û a ÛÛÒ 2.

15 2a. Î ÎÖ Í Ø ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) E ÛÎ Ì (edge) {a,b} E ÛÓ a Ó b Ñ (adjacent) Ú Ë a Ó b Ì ab(= {a,b}) Õ (incident) Ú Ë Í a Ú aa ËÈÎ (loop) Í v Ñ ÕÚ ËÌÛÒ ÛÒ (degree) ÛÒ 0 ÛÍ ÔÌÍ (isolated point) ÛÒ ÛÍ ÈÍ (end-vertix) Ó È. ÎÌ a Û a ÛÛÒ 2. ÊÛÊ ÌÛØ ÈÌÛ Ñ Û ÍÛÛÒ Î ËÓÕÒ ÖË.

16 2b. Ø Ó Ë ÈÌ G = (V,E) Ú ÍÛ w = a 0,a,...,a n ÌÛ i < n a i a i+ E ÓÖË ØÛ G ÛØ (walk) Ó È

17 2b. Ø Ó Ë ÈÌ G = (V,E) Ú ÍÛ w = a 0,a,...,a n ÌÛ i < n a i a i+ E ÓÖË ØÛ G ÛØ (walk) Ó È ÌÛ2ÍÛÍ Ø Ñ ËØ ÈÌ Ë (connected) Ë.

18 2b. Ø Ó Ë ÈÌ G = (V,E) Ú ÍÛ w = a 0,a,...,a n ÌÛ i < n a i a i+ E ÓÖË ØÛ G ÛØ (walk) Ó È ÌÛ2ÍÛÍ Ø Ñ ËØ ÈÌ Ë (connected) Ë. Ú ËÖØ ÈÌ G Ë Ö Ø ÈÌ G 2

19 2b. Ø Ó Ë ÈÌ G = (V,E) Ú ÍÛ w = a 0,a,...,a n ÌÛ i < n a i a i+ E ÓÖË ØÛ G ÛØ (walk) Ó È ÌÛ2ÍÛÍ Ø Ñ ËØ ÈÌ Ë (connected) Ë. Ú ËÖØ ÈÌ G Ë Ö Ø ÈÌ G 2 ÛÊ Ø ÈÌ G = (V,E) a a 2 a ÏÈ a 2 ÐÛØ Ñ Ë Ó ÈÐ Ó ËÓ ÒÚÐ Ë. ÊÏ Û, G/ ÛØ G Û Ñ (component) Ó È.

20 2b. Ø Ó Ë ÈÌ G = (V,E) Ú ÍÛ w = a 0,a,...,a n ÌÛ i < n a i a i+ E ÓÖË ØÛ G ÛØ (walk) Ó È ÌÛ2ÍÛÍ Ø Ñ ËØ ÈÌ Ë (connected) Ë. Ú ËÖØ ÈÌ G Ë Ö Ø ÈÌ G 2 ÛÊ Ø ÈÌ G = (V,E) a a 2 a ÏÈ a 2 ÐÛØ Ñ Ë Ó ÈÐ Ó ËÓ ÒÚÐ Ë. ÊÏ Û, G/ ÛØ G Û Ñ (component) Ó È. Ñ Û G, G 2 Û Ñ Ì

21 2c. ÒÑ ÈÌ ÊÏ Ø ÈÌ G = (V,E 2 ), G 2 = (V 2,E 2 ) V V 2, E E 2 ÛÓ, G G 2 ÛÒÑ ÈÌ (subgraph) ËÓ È. Ñ ÛÛ ÈÌ G ÛÒÑ ÈÌ Ì a b c ÊÏ ÈÌ G ÓÌÛ F G ÏÈ F ÒÒÍËÌ Ê Ì ÈÌ G F Ó Ë. ÍÛ S, G ÏÈ S ÒÒÍË ÍÓÉÍ Õ ËÌ Ñ Ê Ì ÈÌ G S Ó Ë. Ñ Û G, F = {ab}, S = {c}, G F,G S Ì

22 2c. ÒÑ ÈÌ ÊÏ Ø ÈÌ G = (V,E 2 ), G 2 = (V 2,E 2 ) V V 2, E E 2 ÛÓ, G G 2 ÛÒÑ ÈÌ (subgraph) ËÓ È. Ñ ÛÛ ÈÌ G ÛÒÑ ÈÌ Ì a b c ÊÏ ÈÌ G ÓÌÛ F G ÏÈ F ÒÒÍËÌ Ê Ì ÈÌ G F Ó Ë. ÍÛ S, G ÏÈ S ÒÒÍË ÍÓÉÍ Õ ËÌ Ñ Ê Ì ÈÌ G S Ó Ë. Ñ Û G, F = {ab}, S = {c}, G F,G S Ì ÊÏ Ø ÈÌ G = (V,E 2 ), G 2 = (V 2,E 2 ) Û Ð (union) G G 2 (V V 2,E E 2 ) Ê Ë.

23 2d. ÒÌ ÊÏ Ø ÈÌ G = (V,E ), G 2 = (V 2,E 2 ) f : V V 2 Ñ ÖÒÚ ÌÛ Í v,v 2 V ÈÛ Ñ Ì Ó, G Ó G 2 ÒÌ (isomorphic), f ÒÌÙ (isomorphism) Ó È. v v 2 E f(v )f(v 2 ) E 2. Ú ÈÛ Û ÈÌ ÒÌ Ë

24 2d. ÒÌ ÊÏ Ø ÈÌ G = (V,E ), G 2 = (V 2,E 2 ) f : V V 2 Ñ ÖÒÚ ÌÛ Í v,v 2 V ÈÛ Ñ Ì Ó, G Ó G 2 ÒÌ (isomorphic), f ÒÌÙ (isomorphism) Ó È. v v 2 E f(v )f(v 2 ) E 2. Ú ÈÛ Û ÈÌ ÒÌ Ë Ñ ÛÛ ÈÌÛÒÌÙ Ì

25 2e. Î ÎÖ ÈÌ Ö ÈÌ ÔÛ 2 ÍØÌ Ë ÍË.

26 2e. Î ÎÖ ÈÌ Ö ÈÌ ÔÛ 2 ÍØÌ Ë ÍË. Ô ÈÌ ÌÑÖ.

27 2e. Î ÎÖ ÈÌ Ö ÈÌ ÔÛ 2 ÍØÌ Ë ÍË. Ô ÈÌ ÌÑÖ. ØÉ ÈÌ ÔÛÍÛÛÒØÙÚ.

28 2e. Î ÎÖ ÈÌ Ö ÈÌ ÔÛ 2 ÍØÌ Ë ÍË. Ô ÈÌ ÌÑÖ. ØÉ ÈÌ ÔÛÍÛÛÒØÙÚ. Ò ÈÌ ÍÛ Ñ ÛÔ Ð Ö V, V 2 ÛÐ Ö, Ñ ÛÌ Ñ V Ó V 2 Û Í ËÍØÛ Ë.

29 3. ÈÌ Ñ ÈÌ (directed graph) Ó V Ó E V V ÌËÈÍË. ÊÏ ÈÌ G = (V,E) Ú, ÈÌ (underlying graph) G = (V,E ) Ó Ø ÈÌ ÛÛ Ñ Ì ØÛ Ë. V = V Ï (a,b) E ab E Ú ÈÌÓ ÈÌ 5 6

30 3a. Î ÎÖ Í ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) Ó È.

31 3a. Î ÎÖ Í ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) Ó È. E ÛÎ Ì (edge), ÒÌ Ù (arc) Ó È.

32 3a. Î ÎÖ Í ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) Ó È. E ÛÎ Ì (edge), ÒÌ Ù (arc) Ó È. (a,b) E ÛÓ a Ó b Ñ (adjacent) Ú Ë. a Ó b Ì ab(= (a,b)) Õ (incident) Ú Ë.

33 3a. Î ÎÖ Í ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) Ó È. E ÛÎ Ì (edge), ÒÌ Ù (arc) Ó È. (a,b) E ÛÓ a Ó b Ñ (adjacent) Ú Ë. a Ó b Ì ab(= (a,b)) Õ (incident) Ú Ë. Í a Ú aa ËÈÎ (loop) Ó È.

34 3a. Î ÎÖ Í ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) Ó È. E ÛÎ Ì (edge), ÒÌ Ù (arc) Ó È. (a,b) E ÛÓ a Ó b Ñ (adjacent) Ú Ë. a Ó b Ì ab(= (a,b)) Õ (incident) Ú Ë. Í a Ú aa ËÈÎ (loop) Ó È. ÍÛ w = a 0,a,...,a n ÌÛ i < n a i a i+ E ÓÖË ØÛ G ÛØ (walk) Ó È (a 0 a a n Ó ). ÈÌÑ ËÖ ÈÌ Ë (connected) Ë.

35 3a. Î ÎÖ Í ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) Ó È. E ÛÎ Ì (edge), ÒÌ Ù (arc) Ó È. (a,b) E ÛÓ a Ó b Ñ (adjacent) Ú Ë. a Ó b Ì ab(= (a,b)) Õ (incident) Ú Ë. Í a Ú aa ËÈÎ (loop) Ó È. ÍÛ w = a 0,a,...,a n ÌÛ i < n a i a i+ E ÓÖË ØÛ G ÛØ (walk) Ó È (a 0 a a n Ó ). ÈÌÑ ËÖ ÈÌ Ë (connected) Ë. ÌÛ 2 ÍÛÍ Ø Ñ ËÓ Ë (strongly connected) Ë.

36 3a. Î ÎÖ Í ÈÌ G = (V,E) Ú V ÛÎ Í (vertix), ÒÌ Í, ÛÈÔ (node) Ó È. E ÛÎ Ì (edge), ÒÌ Ù (arc) Ó È. (a,b) E ÛÓ a Ó b Ñ (adjacent) Ú Ë. a Ó b Ì ab(= (a,b)) Õ (incident) Ú Ë. Í a Ú aa ËÈÎ (loop) Ó È. ÍÛ w = a 0,a,...,a n ÌÛ i < n a i a i+ E ÓÖË ØÛ G ÛØ (walk) Ó È (a 0 a a n Ó ). ÈÌÑ ËÖ ÈÌ Ë (connected) Ë. ÌÛ 2 ÍÛÍ Ø Ñ ËÓ Ë (strongly connected) Ë. Ñ ËÍÑ Ë Ö ÈÌ Ò.

37 3b. ÒÌ ÊÏ ÈÌ G = (V,V 2 ), G 2 = (V 2,E 2 ) f : V V 2 Ñ ÖÒÚ ÌÛ Í v,v 2 V ÈÛ Ñ Ì Ó, f G Ó G 2 ÒÌ (isomorphic), f ÒÌÙ (isomorphism) Ó È. v v 2 E f(v )f(v 2 ) E 2.

38 3b. ÒÌ ÊÏ ÈÌ G = (V,V 2 ), G 2 = (V 2,E 2 ) f : V V 2 Ñ ÖÒÚ ÌÛ Í v,v 2 V ÈÛ Ñ Ì Ó, f G Ó G 2 ÒÌ (isomorphic), f ÒÌÙ (isomorphism) Ó È. v v 2 E f(v )f(v 2 ) E 2. Ú ÈÛ Û ÈÌ ÒÌ Ë. 5 6

39 3b. ÒÌ ÊÏ ÈÌ G = (V,V 2 ), G 2 = (V 2,E 2 ) f : V V 2 Ñ ÖÒÚ ÌÛ Í v,v 2 V ÈÛ Ñ Ì Ó, f G Ó G 2 ÒÌ (isomorphic), f ÒÌÙ (isomorphism) Ó È. v v 2 E f(v )f(v 2 ) E 2. Ú ÈÛ Û ÈÌ ÒÌ Ë. 5 6 Ñ ÛÛ Û ÈÌ ÒÌÏÌ 5 6

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