The nth detour chromatic number, χn(G) of a graph G is the minimum number of colours required to colour the vertices of G such that no path with more than n vertices is monocoloured. The number of vertices in a longest path of G is denoted by τ (G) . We conjecture that χn(G) ≤ d τ(G) n e for every graph G and every n ≥ 1 and we prove results that support the conjecture. We also present some suf...

The 7-axis articulated robot can arbitrarily rotate the position of elbow by changing angle 7th axis. When a makes detour, it is necessary to properly control both and orientation tool E-axis angle. In this study, we propose deep reinforcement learning method for automatically generating detour paths including

Let be any connected graph with vertices set and edges . For two distinct , the detour distance between which is denoted by a longest path in The polynomial of ; defined In this paper, theta uniform will computed.

This study presents an integrated model and its solution algorithm for freeway corridor control during incident management. With a parallel arterial as the detour route, the proposed model aims to produce the optimal diversion rates from the freeway mainline to relieve the congestion at the incident segment, and concurrently adjust signal timings at the arterial intersections to best accommodat...

We consider the following natural “above guarantee” parameterization of the classical Longest Path problem: For given vertices s and t of a graph G, and an integer k, the problem Longest Detour asks for an (s, t)-path in G that is at least k longer than a shortest (s, t)-path. Using insights into structural graph theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on undirec...

In 1982 Laborde, Payan and Xuong [Independent sets and longest directed paths in digraphs, in: Graphs and other combinatorial topics (Prague, 1982) 173–177 (Teubner-Texte Math., 59 1983)] conjectured that every digraph has an independent detour transversal (IDT), i.e. an independent set which intersects every longest path. Havet [Stable set meeting every longest path, Discrete Math. 289 (2004) ...

A spanning subgraph H of a graph G is a 2-detour subgraph of G if for each x, y ∈ V (G), dH(x, y) ≤ dG(x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2-detour subgraph of the n-dimensional hypercube Qn has at least c log2 n · 2n edges.

This paper presents a generalized diversion control model for freeway incident management that is capable of concurrently optimizing the detour rates and arterial signal timings over multiple roadway corridor segments between the freeway and its neighboring arterial. To capture various operational complexities due to the interactions between multiple diversions, this study has extended the mode...

The Detour framework is a software infrastructure for alternative routing on the internet. It is being developed in conjunction with the Detour routing project, to serve as a mechanism for experimental deployment of new, performance-sensitive routing algorithms in the wide-area. The framework implements a virtual network that allows users of our system to take advantage of improved routing for ...