Firstly, for a graph weighted in a bounded incline algebra (or called a dioid), a longest path problem (LPP, for short) is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and the most reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximu...

Let G be the chemical graph of a molecule. The matrix D = [dij ] is called the detour matrix of G, if dij is the length of longest path between atoms i and j. The sum of all entries above the main diagonal of D is called the detour index of G. In this paper, a new algorithm for computing the detour index of molecular graphs is presented. We apply our algorithm on copper and silver nanoclusters ...

Given an undirected graph G=(V,E), the longest induced path problem (LIPP) consists of obtaining a maximum cardinality subset W⊆V such that W induces simple in G. In this paper, we propose two new formulations with exponential number constraints for problem, together effective branch-and-cut procedures its solution. While first formulation (cec) is based on explicitly eliminate cycles, second o...

The longest path problem asks for a path with the largest number of vertices in a given graph. In contrast to the Hamiltonian path problem, until recently polynomial algorithms for the longest path problem were known only for small graph classes, such as trees. Recently, a polynomial algorithm for this problem on interval graphs has been presented in Ioannidou et al. (2011) [19] with running ti...

The longest path problem asks for a path with the largest number of vertices in a given graph. In contrast to the Hamiltonian path problem, until recently polynomial algorithms for the longest path problem were known only for small graph classes, such as trees. Recently, a polynomial algorithm for this problem on interval graphs has been presented in [20] with running time O(n) on a graph with ...