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The Longest Path Problem is a question of finding the maximum length betweenpairs of vertices of a graph. In the general case, the problem is NP-hard.However, there is a small collection of graph classes for which there exists anefficient solution. Current approaches involve either approximation orcomputational enumeration. For Tree-like classes of graphs, there areapproximation and enumeration algorithms which solves the problem efficiently.We propose a new method of approaching the longest path problem with algebraicoperations and conditions that exactly identify and or approximate the solutionin polynomial time. We next introduce a 'booleanize' mapping on the adjacencymatrix of a graph which we prove identifies the solution for trees, uniformblock graphs, block graphs, and directed acyclic graphs, with approachedconditions. Finally, we display the algorithms to find the solution, inaddition to algorithms that generate all the longest paths.

An Algebraic Approach to the Longest Path Problem