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A tour of a graph (digraph or sequential machine) is a sequence of nodes from the graph such that each node appears at least once and two nodes are adjacent in the sequence only if they are adjacent in the graph. Finding the shortest tour of a graph is known to be an NP-complete problem. Several theorems are given that show that there are classes ofgraphs in which the shortest tour can be found easily. For more general graphs, we present approximating algorithms for finding short tours. For undirected graphs, the approximating algorithms give tours at worst a constant times the length of the shortest tour. For directed graphs, the size of the calculated tour is bounded by the size of the digraph times the shortest tour. Not only are the bounds worse for the directed case, but the running times of the approximating algorithms are also larger than those for the undirected case.