We present several results concerning the Laborde-Payan-Xuang conjecture stating that in every digraph there exists an independent set of vertices intersecting every longest path. The digraphs we consider are defined in terms of local semicompleteness and local transitivity. We also look at oriented graphs for which the length of a longest path does not exceed 4.

The longest induced path problem consists in finding a maximum subset of vertices graph such that it induces simple path. We propose new exact enumerative algorithm solves problems with up to 138 and 493 edges heuristic for larger problems. Detailed computational experiments compare the results obtained by algorithms other approaches literature investigate characteristics optimal solutions.

Five experiments examined the organization of spatial memory of an irregular path learned by walking with vision. Two hypotheses were tested: (a) that participants would establish a single global frame of reference to organize the spatial memory of the path; or alternatively (b) that participants would establish path-aligned reference directions at each path leg but not establish a global frame...

A chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path  $P$ is called a monophonic path if it is a chordless path. A longest $x-y$ monophonic path is called an $x-y$ detour monophonic path. A  detour monophonic graphoidal cover of a graph $G$ is a collection $psi_{dm}$ of detour monophonic paths in $G$ such that every vertex of $G$ is an internal vertex  of at most on...

We provide conditions on the minimum degree, the girth and the length of a longest path for a digraph to have the following property: Every longest path intersects every maximal independent set. Digraphs with this property satisfy, in a strong sense, the Laborde-PayanXuong conjecture (see [10]).

In this paper, the longest Hamiltonian cycle problem and the longest Hamiltonian path problem are proved to be NPO-complete. @ 1998 Published by Elsevier Science B.V.

The 2-Disjoint Connected Subgraphs problem asks if a given graph has two vertex-disjoint connected subgraphs containing prespecified sets of vertices. We show that this problem is NP-complete even if one of the sets has cardinality 2. The Longest Path Contractibility problem asks for the largest integer ` for which an input graph can be contracted to the path P` on ` vertices. We show that the ...

We study the problem of finding small integer arc lengths in a digraph such that some given paths are uniquely determined shortest paths. This Inverse Shortest Paths (ISP) problem arises in the planning of shortest path networks. We present an O(|V |)-approximation algorithm for ISP and show that minimizing the longest arc or the longest path length is APX -hard. Thus, deciding if a given path ...

Many known algorithms are based on selection in a set whose cardinality is superlinear in terms of the input length. It is desirable in these cases to have selection algorithms that run in sublinear time in terms of the cardinality of the set. This paper presents a successful development in this direction. The methods developed here are applied to improve the previously known upper bounds for t...