hw9solns - Discrete Mathematics Spring 2004 Homework 9...

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Discrete Mathematics, Spring 2004 Homework 9 Sample Solutions 7.1 #36. A vertex v in a tree T is a center for T if the eccentricity of v is minimal; that is, if the maximum length of a simple path starting from v is less than or equal to the maximum length of a simple path starting from any other vertex w . Show that a tree has either one or two centers. Solution . Let P be a simple path of maximal length in T . (Such a path surely exists; in fact, we have seen this construction before, when establishing equivalent characterizations of trees.) The length of this path is either even or odd. In the first case, denote this length by 2 m ; then we have P = ( v 0 , v 1 , . . . , v m 1 , v m , v m +1 , . . . , v 2 m 1 , v 2 m ) , where all of the v i are distinct. We claim that v m is the unique center for T . To see this, notice first that the eccentricity of v m is actually equal to m (clearly it must be at least m ). If it were not, then there would exist a simple path ( v m = w 0 , w 1 , . . . , w k = w ) with k > m . Since trees are acyclic, there are only two possibilities: w j = v m + j for all j some nonnegative integer j 0 , and w j / ∈ { v 0 , . . . , v 2 m } for all j > j 0 , or w j = v m j for all j j 0 , and w j / ∈ { v 0 , . . . , v 2 m } for all j > j 0 . In the first case, the path P = ( v 0 , v 1 , . . . , v m = w 0 , w 1 , . . . , w k = w ) would be a simple path of length greater than that of P , contradicting the way in which we chose P . In the second case, the path P ′′ = ( w = w k , . . . , w 1 , w 0 = v m , v m 1 , . . . , v 0 ) would be a simple path of length greater than that of P , again a contradiction. Hence the eccentricity of v m is equal to m . Now consider any of the remaining vertices u in T . If u is one of the v i , then its eccentricity must be > m , for if u = v i with i m then ( v i , v i +1 , . . . , v 2 m ) is a simple path of length > m , and if u = v i with i > m then ( v i , v i 1 , . . . , v 0 ) is a simple path of length > m . If u is not one of the v i
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