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Digraphs+Theory,+Algorithms+and+Applications_Part16

# Digraphs+Theory,+Algorithms+and+Applications_Part16 - 6.1...

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6.1 Hamiltonian Paths with a Prescribed End-Vertex 283 Theorem 6.1.2 [66] Let D = ( V, A ) be a digraph which is either semicom- plete bipartite or extended locally out-semicomplete and let x V . Then D has a hamiltonian path starting at x if and only if D contains a 1-path-cycle factor F of D such that the path of F starts at x , and, for every vertex y of V -{ x } , there is an ( x, y ) -path 1 in D . Moreover, if D has a hamiltonian path starting at x , then, given a 1-path-cycle factor F of D such that the path of F starts at x , the desired hamiltonian path can be found in time O ( n 2 ) . Proof: As the necessity is clear, we will only prove the sufficiency. Suppose that F = P C 1 ... C t is a 1-path-cycle factor of D that consists of a path P starting at x and cycles C i , i = 1 , ..., t . Suppose also that every vertex of D is reachable from x . Then, without loss of generality, there is a vertex of P that dominates a vertex of C 1 . Let P = x 1 x 2 . . . x p , C 1 = y 1 y 2 . . . y q y 1 , where x = x 1 and x k y s for some k ∈ { 1 , 2 , . . . , p } , s ∈ { 1 , 2 , . . . , q } . We show how to find a new path starting at x which contains all the vertices of V ( P ) V ( C 1 ). Repeating this process we obtain the desired path. Clearly, we may assume that k < p and that x p has no arc to V ( C 1 ). Assume first that D is an extended locally out-semicomplete digraph. If P has a vertex x i which is similar to a vertex y j in C 1 , then x i y j +1 , y j x i +1 A and using these arcs we see that P [ x 1 , x i ] C [ y j +1 , y j ] P [ x i +1 , x p ] is a path starting from x and containing all the vertices of P C 1 . If P has no vertex that is similar to a vertex in C 1 , then we can apply the result of Exercise 4.37 to P [ x k , x p ] and x k C 1 [ y s , y s - 1 ] and merge these two paths into a path R starting from x k and containing all the vertices of P [ x k , x p ] C 1 . Now, P [ x 1 , x k - 1 ] R is a path starting at x and containing all the vertices of P C 1 . Suppose now that D is semicomplete bipartite. Then either y s - 1 x k +1 , which implies that P [ x 1 , x k ] C 1 [ y s , y s - 1 ] P [ x k +1 , x p ] is a path starting at x and covering all the vertices of P C 1 , or x k +1 y s - 1 . In the latter case, we consider the arc between x k +2 and y s - 2 . If y s - 2 x k +2 we can construct the desired path, otherwise we continue to consider arcs between x k +3 and y s - 3 and so on. If we do not construct the desired path in this way, then we find that the last vertex of P dominates a vertex in C 1 , contradicting our assumption above. Using the process above and breadth-first search, one can construct an O ( n 2 )-algorithm for finding the desired hamiltonian path starting at x . ut Just as the problem of finding a minimum path factor generalizes the hamiltonian path problem, we may generalize the problem of finding a hamil- tonian path starting at a certain vertex to the problem of finding a path factor with as few paths as possible such that one of these paths starts at a specified vertex x . We say that a path factor starts at x if one of its paths starts at x and denote by pc x ( D ) the minimum number of paths in a path factor that 1

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