Discrete Mathematics with Graph Theory (3rd Edition) 298

Discrete Mathematics with Graph Theory (3rd Edition) 298 -...

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296 Solutions to Exercises Partition into pairs Sum of lengths of shortest paths Graph (a) Graph (b) Graph (c) {A,B}, {C,D}, {E,F} 1+1+1=3 2+2+3=7 5 + 2 + 5 = 12 {A, B}, {C, E}, {D, F} 1+2+2=5 2+2+4=8 5 + 5 + 6 = 16 {A,B}, {C, F}, {D, E} 1+3+1=5 2+3+2=7 5 + 8+3 = 16 {A,C}, {B,D}, {E,F} 2+2+1=5 3+4+3=10 6+6+5=17 {A,C}, {B,E}, {D,F} 2+3+2=7 3+3+4=10 6 + 6 +6 = 18 {A, C}, {B,F}, {D,E} 2+2+1=5 3+2+2=7 6+7+3 = 16 {A,D}, {B,C}, {E,F} 3+1+1=5 4+3+3=10 4+4+ 5 = 13 {A,D}, {B,E}, {C,F} 3+3+3=9 4+3+3=10 4+6+8 = 18 {A, D}, {B, F}, {C, E} 3+2+2=7 4+2+2=8 4+ 7+ 5 = 16 {A, E}, {B, C}, {D, F} 2+1+2=5 3+3+4=10 3 +4 + 6 = 13 {A, E}, {B, D}, {C, F} 2+2+3=7 3+4+3=10 3+6+8=17 {A,E}, {B,F}, {C,D} 2+2+1=5 3+2+2=7 3 + 7 + 2 = 12 {A,F}, {B, C}, {D,E} 1+1+1=3 2+3+2=7 4+4+3=11 {A,F}, {B,D}, {C,E} 1+2+2=5 2+4+2=8 4+ 6 +5 = 15 {A, F}, {B, E}, {C, D} 1+3+1=5 2+3+2=7 4+ 6+2 = 12 Graph (d) Partition into pairs Sum of lengths of shortest paths {A,B}, {C,D}
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Unformatted text preview: {A, C}, {B, D} {A,D},{B,C} 5+3=8 4+6=10 3+6=9 6. [BB] The shortest route from X to X is a circuit (perhaps in a pseudograph) passing through Y. Hence, it can also be viewed as a circuit from Y to Y. Any other route from Y to Y could also be viewed as a route from X to X. So there cannot be any shorter route from Y to Y. 7. The six vertices of odd degree are {A, B, C, D, E, F} and the shortest path totals for all partitions are given. {A, B}, {C, D}, {E, F} {A, B}, {C, F}, {D, E} {A, C}, {B, E}, {D, F} {A, D}, {B, C}, {E, F} {A, D}, {B, F}, {C, E} {A, E}, {B, D}, {C, F} {A, F}, {B, C}, {D, E} {A, F}, {B, E}, {C, D} 2+3+1=6 2+3+1=6 3+3+2=8 4+3+1=8 4+4+3=11 5+2+3=10 6+3+1=10 6+3+3 = 12 {AB,CE,DF} {AC,BD,EF} {AC,BF,DE} {AD,BE,CF} {AE,BC,DF} {AE,BF,CD} {AF,BD,CE} 2+3+2=7 3+2+1=6 3+4+1=8 4+3+3=10 5+ 3 +2 = 10 5+4+ 3 = 12 6+2+3=11 So we see that one solution is to add cBPies of edges AX, X B, C M, M N, N D and EF, as shown. D A...
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