CPS 420 Euler Circuit Construction Sophie Quigley Original

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CPS 420 Euler Circuit Construction © Sophie Quigley

CPS 420 Euler Circuit Construction © Sophie Quigley

Original Graph G V(G) = {A, B, C, D, E, F, G, H, I,

Original Graph G V(G) = {A, B, C, D, E, F, G, H, I, J} E(G)={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}

Graph G’ Ø Let G’ = G E(G’)= {1, 2, 3, 4, 5, 6,

Graph G’ Ø Let G’ = G E(G’)= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} V(G’) = {A, B, C, D, E, F, G, H, I, J}

Graph G’ Ø Pick a vertex of G’: v=B E(G’)= {1, 2, 3, 4,

Graph G’ Ø Pick a vertex of G’: v=B E(G’)= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} V(G’) = {A, B, C, D, E, F, G, H, I, J}

Circuit C=C’ Ø Let C = C’ = a circuit in G which starts

Circuit C=C’ Ø Let C = C’ = a circuit in G which starts and ends in v = B 2 C 13 I 8 H 12 B E(C) = E(C’) = {2, 13, 8, 12 } V(C) = V(C’) = {B, C, I, H} E(G’)= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} V(G’) = {A, B, C, D, E, F, G, H, I, J}

Iterate on G’ Ø Let G’ be the new graph s. t. E(G’)= E(G’)

Iterate on G’ Ø Let G’ be the new graph s. t. E(G’)= E(G’) – E(C’) = {1, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 16, 17 } V(G’) = V(G’) – {all isolated vertices once edges in E(C’) have been removed} = {A, B, C, D, E, F, G, H, I, J} C = C’= B 2 C 13 I 8 H 12 B E(C) = E(C’) = {2, 13, 8, 12 } V(C) = V(C’) = {B, C, I, H}

Iterate on C’ Ø Pick a vertex w from V(C) V(G’) Ø w=H E(G’)=

Iterate on C’ Ø Pick a vertex w from V(C) V(G’) Ø w=H E(G’)= {1, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 16, 17 } V(G’) = {A, B, C, D, E, F, G, H, I, J} C = B 2 C 13 I 8 H 12 B E(C) = {2, 13, 8, 12 } V(C) = {B, C, I, H}

Iterate on C’ Ø Let C’ = circuit of G’ starting at w Ø

Iterate on C’ Ø Let C’ = circuit of G’ starting at w Ø C’ = H 15 E 5 F 6 G 7 H Ø Integrate C’ into C: C = B 2 C 13 I 8 H 15 E 5 F 6 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7 } V(C) = {B, C, I, H, E, F, G}

Iterate on G’ Ø Let G’ be the new graph s. t. E(G’)= E(G’)

Iterate on G’ Ø Let G’ be the new graph s. t. E(G’)= E(G’) – E(C’) = {1, 3, 4, 9, 10, 11, 14, 16, 17 } V(G’) = V(G’) – {all isolated vertices once edges in E(C’) have been removed} = {A, B, C, D, E, G, I, J} C = B 2 C 13 I 8 H 15 E 5 F 6 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7 } V(C) = {B, C, I, H, E, F, G}

Iterate on C’ Ø Pick a vertex w from V(C) V(G’) Ø w=I E(G’)=

Iterate on C’ Ø Pick a vertex w from V(C) V(G’) Ø w=I E(G’)= {1, 3, 4, 9, 10, 11, 14, 16, 17 } V(G’) = {A, B, C, D, E, G, I, J} C = B 2 C 13 I 8 H 15 E 5 F 6 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7 } V(C) = {B, C, I, H, E, F, G}

Iterate on C’ Ø Let C’ = circuit of G’ starting at w Ø

Iterate on C’ Ø Let C’ = circuit of G’ starting at w Ø C’ = I 11 B 1 A 10 J 9 I Ø Integrate C’ into C: C = B 2 C 13 I 11 B 1 A 10 J 9 I 8 H 15 E 5 F 6 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7, 11, 1, 10, 9 } V(C) = {B, C, I, H, E, F, G, J, A}

Iterate on G’ Ø Let G’ be the new graph s. t. E(G’)= E(G’)

Iterate on G’ Ø Let G’ be the new graph s. t. E(G’)= E(G’) – E(C’) = {3, 4, 16, 17 } V(G’) = V(G’) – {all isolated vertices once edges in E(C’) have been removed} = {C, D, E, I, J} C = B 2 C 13 I 11 B 1 A 10 J 9 I 8 H 15 E 5 F 6 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7, 11, 1, 10, 9 } V(C) = {B, C, I, H, E, F, G, J, A}

Iterate on C’ Ø Pick a vertex w from V(C) V(G’) Ø w=G E(G’)=

Iterate on C’ Ø Pick a vertex w from V(C) V(G’) Ø w=G E(G’)= {3, 4, 16, 17 } V(G’) = {C, D, E, I, J} C = B 2 C 13 I 11 B 1 A 10 J 9 I 8 H 15 E 5 F 6 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7, 11, 1, 10, 9 } V(C) = {B, C, I, H, E, F, G, J, A}

Iterate on C’ Ø Let C’ = circuit of G’ starting at w Ø

Iterate on C’ Ø Let C’ = circuit of G’ starting at w Ø C’ = G 16 I 17 G Ø Integrate C’ into C: C = B 2 C 13 I 11 B 1 A 10 J 9 I 8 H 15 E 5 F 6 G 16 I 17 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7, 11, 1, 10, 9 , 16, 17} V(C) = {B, C, I, H, E, F, G, J, A}

Iterate on G’ Ø Let G’ be the new graph s. t. E(G’)= E(G’)

Iterate on G’ Ø Let G’ be the new graph s. t. E(G’)= E(G’) – E(C’) = {3, 4, 14} V(G’) = V(G’) – {all isolated vertices once edges in E(C’) have been removed} = {C, D, E} C = B 2 C 13 I 11 B 1 A 10 J 9 I 8 H 15 E 5 F 6 G 16 I 17 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7, 11, 1, 10, 9 , 16, 17} V(C) = {B, C, I, H, E, F, G, J, A}

Iterate on C’ Ø Pick a vertex w from V(C) V(G’) Ø w=C E(G’)=

Iterate on C’ Ø Pick a vertex w from V(C) V(G’) Ø w=C E(G’)= {3, 4, 14} V(G’) = {C, D, E} C = B 2 C 13 I 11 B 1 A 10 J 9 I 8 H 15 E 5 F 6 G 16 I 17 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7, 11, 1, 10, 9 , 16, 17} V(C) = {B, C, I, H, E, F, G, J, A}

Iterate on C’ Ø Let C’ = circuit of G’ starting at w Ø

Iterate on C’ Ø Let C’ = circuit of G’ starting at w Ø C’ = C 3 D 4 E 14 C Ø Integrate C’ into C: C = B 2 C 3 D 4 E 14 C 13 I 11 B 1 A 10 J 9 I 8 H 15 E 5 F 6 G 16 I 17 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7, 11, 1, 10, 9, 16, 17, 3, 4, 14} V(C) = {B, C, I, H, E, F, G, J, A, D}

Iterate on G’ Ø Let G’ be the new graph s. t. E(G’)= E(G’)

Iterate on G’ Ø Let G’ be the new graph s. t. E(G’)= E(G’) – E(C’) = V(G’) = V(G’) – {all isolated vertices once edges in E(C’) have been removed} Ø V(G’) = C = B 2 C 3 D 4 E 14 C 13 I 11 B 1 A 10 J 9 I 8 H 15 E 5 F 6 G 16 I 17 G 7 H 12 B E(C) = {2, 13, 8, 12, 15, 5, 6, 7, 11, 1, 10, 9, 16, 17, 3, 4, 14} V(C) = {B, C, I, H, E, F, G, J, A, D}

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