Exam 10: Graphs

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$K _ { 5 }$ is nonplanar. $K _ { 2,3 }$ is planar, as can easily be seen by drawing it with no crossings, or since it has no subgraph homeomorphic to $K _ { 3.3 }$ or $K _ { 5 } . W _ { 5 }$ is planar as is seen from the usual way of drawing it.

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First iteration: distinguished vertices: $a$ ; labels: $a : 0 , b : 3 , c : 7 , d : \infty , e : \infty , z : \infty$ . Second iteration: distinguished vertices: $a , b ;$ labels: $a : 0 , b : 3 , c : 5 , d : 9 , e : \infty , z : \infty$ . Third iteration: distinguished vertices: $a , b , c ;$ labels: $a : 0 , b : 3 , c : 5 , d : 6 , e : 11 , z : \infty$ . Fourth iteration: distinguished vertices: $a , b , c , d ;$ labels: $a : 0 , b : 3 , c : 5 , d : 6 , e : 8 , z : 14$ . Fifth iteration: distinguished vertices: $a , b , c , d , e ;$ labels: $a : 0 , b : 3 , c : 5 , d : 6 , e : 8 , z : 13$ . Since at the next iteration $z$ is a distinguished vertex, we conclude that the shortest path has length 13 .

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The graph is bipartite. The vertex set can be partitioned into {a, c, e} and {b, d, f}. There are no edges connecting a vertex in one set and a vertex in the other set.