Efficient Algorithms for the Longest Path Problem

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Abstract

The longest path problem is to find a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, very few graph classes are known where the longest path problem can be solved efficiently. For a tree, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and it then solves the longest path problem efficiently for weighted trees, block graphs, ptolemaic graphs, and cacti. We next propose three new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on those classes. As a corollary, it is also shown that the problem can be solved efficiently on threshold graphs.