The longest path problem asks to find a path of maximum length in a given graph. The problem is NP-complete, but there exists an efficient dynamic programming solution for acyclic digraphs.
The detour matrix is a real symmetric matrix whose 
th entry is the length of the longest
path from vertex 
to vertex 
.
For a traceable graph, longest paths correspond to Hamiltonian paths.
Finding a longest path is challenging for stacked book graphs and Apollonian networks.
See also
Antipodal Graph, Bellman-Ford Algorithm, Detour Index, Detour Matrix, Detour Polynomial, Graph Circumference, Hamiltonian Cycle, Hamiltonian Path, Path Polynomial, Shortest Path Problem, Traveling Salesman Problem
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References
Zamfirescu, T. "On Longest Paths and Circuits in Graphs." Math. Scand. 38, 211-239, 1976.
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Cite this as:
Weisstein, Eric W. "Longest Path." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LongestPath.html