We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A directed cycle is called quasi-monochromatic if with at most one exception all of its arcs are coloured alike. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the following t...

We investigate the hardness of approximating the longest path and the longest cycle in directed graphs on n vertices. We show that neither of these two problems can be polynomial time approximated within n1-ε for any ε > 0 unless P = NP. In particular, the result holds for digraphs of constant bounded outdegree that contain a Hamiltonian cycle. Assuming the stronger complexity conjecture that S...

Tewes and Volkmann [Australas. J. Combin. 18 (1998), 293–301] proved that for an integer k ≥ 3, every in-tournament of minimum in-degree d ≥ 1 and of order at most kd contains a directed cycle of length at most k. In other words, they proved that the Caccetta-Häggkvist conjecture, with respect to the minimum in-degree, is true for in-tournaments. In the same paper, they proved also that every s...

Given a directed graph, an acyclic set is a set of vertices inducing a directed subgraph with no directed cycle. In this note we show that for all integers n ≥ g ≥ 3, there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most dn(g−2)+1 g−1 e. When g = 3 this result disproves a conjecture of Harutyunyan and shows that a question of Albert...

Fine-grained reductions have established equivalences between many core problems with Õ(n3)-time algorithms on n-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also have Õ(mn)-time algorithms on m-edge n-node weighted graphs, and such algorithms have wider applicability. Are these mn boun...

We consider the problem of computing a minimum cycle basis in a directed graph. The input to this problem is a directed graph G whose edges have nonnegative weights. A cycle in this graph is actually a cycle in the underlying undirected graph with edges traversable in both directions. A {−1, 0, 1} edge incidence vector is associated with each cycle: edges traversed by the cycle in the right dir...

Let G be an undirected 2-edge connected graph with nonnegative edge weights and a distinguished vertex z. For every node consider a shortest cycle containing this node and z in G. The cycle-radius of G is the maximum length of a cycle in this set. Let H be a directed graph obtained by directing the edges of G. The cycle-radius of H is similarly defined except that cycles are replaced by directe...

In a directed graph with edge weights, the mean weight of a directed cycle is the weight of its edges divided by their number. The minimum cycle mean of the graph is the minimum mean weight of a cycle. Karp gave a characterization of minimum cycle mean and an O(nm) algorithm to compute it, where n is the number of vertices and m is the number of edges. However, an algorithm he suggested for ide...

objective(s):multidrug resistance (mdr) of cancer cells is a major obstacle to successful chemotherapy. overexpression of breast cancer resistance protein (bcrp) is one of the major causes of mdr. in addition, it has been shown that pi3k/akt signaling pathway involves in drug resistance. therefore, we evaluated the effects of novel approaches including sirna directed against bcrp and targeted t...

Algorithm tests if a Hamiltonian cycle exists in directed graphs, if it is exists algorithm can show found Hamiltonian cycle. If you want to test an undirected graph, such a graph should be converted to the form of directed graph. Previously known algorithm solving Hamiltonian cycle problem brute-force search can’t handle relatively small graphs. Algorithm presented here is referred to simply a...