The topological ordering algorithm sorts nodes of a directed graph such that the order of the tail of each arc is lower than the order of its head. In this paper, we introduce the notion of covering between nodes of a directed graph. Then, we apply the topological orderingalgorithm on graphs containing the covering nodes. We show that there exists a cut set withforward arcs in these...

We present two algorithms for maintaining the topological order of a directed acyclic graph with n vertices, under an online edge insertion sequence of m edges. Efficient algorithms for online topological ordering have many applications, including online cycle detection, which is to discover the first edge that introduces a cycle under an arbitrary sequence of edge insertions in a directed grap...

Let G = (V,E) be a directed acyclic graph (dag) with n = |V | and m = |E|. We say that a total ordering ≺ on vertices V is a topological ordering if for every edge (u,v) ∈ E , we have u ≺ v. In this paper, we consider the problem of maintaining a topological ordering subject to dynamic changes to the underlying graph. That is, we begin with an empty graph G =(V, / 0) consisting of n nodes. The ...

this study aims to deal with a more realistic combined problem of project scheduling and material ordering. the goal is to minimize the total material holding and ordering costs by determining the starting time of activities along with material ordering schedules subject to some constraints. the problem is first mathematically modelled. then a hybrid simulated annealing and genetic algorithm is...

many ranking methods have been proposed so far. however, there is yet no method that can always give a satisfactory solution to every situation; some are counterintuitive, not discriminating; some use only the local information of fuzzy values; some produce different ranking for the same situation. for overcoming the above problems, we propose a new method for ranking fuzzy quantities based on ...

We present a simple algorithm which maintains the topological order of a directed acyclic graph with n nodes under an online edge insertion sequence in O(n) time, independent of the number of edges m inserted. For dense DAGs, this is an improvement over the previous best result of O(min{m 2 log n,m 3 2 + n log n}) by Katriel and Bodlaender. We also provide an empirical comparison of our algorit...

We present an Õ(n)-time algorithm for maintaining the topological order of a directed acyclic graph with n vertices while inserting m edges. This is an improvement over the previous result of O(n) by Ajwani, Friedrich, and Meyer.