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- W2034130047 abstract "The class of dynamic graphs g(m, k, N) consists of directed graphs with the following structure: the m ( k + 1) vertices are arranged in k + 1 stages each containing m vertices, and each of the N oriented edges points from stage i to stage i + 1, for some i = 1, 2,..., k. g(m, k, N) is a sub-class of multipartite graphs. There exists exactly M = ( N m2k ) labelled digraphs in the class g(m, k, N) . If we attribute equal probability 1/ M to these M dynamic graphs, we obtain the random dynamic graph G N . The main results of this work are: (a) if N/km ⩽ B < 1, then G N contains no directed k-path (from stage 1 to stage k + 1), with probability approaching unity as m and k approach infinity; (b) if N/km →∞ and k/m ⩽ c < ∞, then there exist many k -paths in G N , with probability approaching unity; (c) the same results hold for the random dynamic graph G p where each potential edge exists with probability p = N/m 2 k; (d) if N = m 2 k ( p = 1), and edge i has random weight C i , i = 1,..., N , we obtain a good upper bound on the value of the minimum weight k-path, and an exact value for the bottleneck problem on k -paths, as m , k → ∞, but k/m ⩽ C < ∞." @default.
- W2034130047 created "2016-06-24" @default.
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- W2034130047 date "1987-07-01" @default.
- W2034130047 modified "2023-09-26" @default.
- W2034130047 title "Maximal Paths in Random Dynamic Graphs" @default.
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- W2034130047 doi "https://doi.org/10.1016/s0195-6698(87)80036-7" @default.
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