A noneven digraph is a directed graph with self-loops such that at least one weight-ing of its arcs by 0 or 1 results in no even cycles. A directed cycle in a weighted digraph is even if the sum of weights of its arcs is an even number. The characterisation of 2-connected undirected graphs G such that their symmetric directed counterparts G are noneven, is obtained via a result of Kastelyn conc...

We study directed random graphs (random graphs whose edges are directed), and present new results on the so-called strong components of those graphs. We provide analytic and simulation results on two special classes of strong component, called cycle components and knots, which are important in random networks that represent certain computational systems.

This paper considers weight-maximizing variants of the classical symmetric and asymmetric travelingsalesman problems. Like their weight-minimizing counterparts, these variants are MAX SNP-hard. We present the first nontrivial approximation algorithms for these problems. Our algorithm for directed graphs finds a tour whose weight is at least 38 63 ≈ 0.603 times the weight of a maximum-weight tou...

This paper considers weight-maximizing variants of the classical symmetric and asymmetric travelingsalesman problems. Like their weight-minimizing counterparts, these variants are M A X SNP-hard. We present the first nontrivial approximation algorithms for these problems. Our algorithm for directed graphs finds a tour whose weight is a t least Q M 0.603 times the weight of a maximum-weight tour...

We give an fpt approximation algorithm for the directed vertex disjoint cycle problem. Given a directed graph G with n vertices and a positive integer k, the algorithm constructs a family of at least k/ρ(k) disjoint cycles of G if the graph G has a family of at least k disjoint cycles (and otherwise may still produce a solution, or just report failure). Here ρ is a computable function such that...

We study the design of capacitated survivable networks using directed p-cycles. A p-cycle is a cycle with at least three arcs, used for rerouting disrupted flow during edge failures. Survivability of the network is accomplished by reserving sufficient slack on directed p-cycles so that if an edge fails, its flow can be rerouted along the p-cycles. We describe a model for designing capacitated s...

For a directed graph G without loops or parallel edges, let β(G) denote the size of the smallest feedback arc set, i.e., the smallest subset X ⊂ E(G) such that G \X has no directed cycles. Let γ(G) be the number of unordered pairs of vertices of G which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least r ≥ 4 satisfies β(G) ≤ cγ(G)/r, where c ...

Given a directed graph G with non-negative cost on the arcs, a directed tour cover T ofG is a cycle (not necessarily simple) inG such that either head or tail (or both of them) of every arc in G is touched by T . The minimum directed tour cover problem (DToCP), which is to find a directed tour cover of minimum cost, is NP -hard. It is thus interesting to design approximation algorithms with per...

Let G be a directed graph and H a subgraph of G. A D(G,H, λ) design is a multiset D of subgraphs of G each isomorphic to H so that every directed 2-path in G lies in exactly λ subgraphs in D. In this paper, we show that there exists a D(K∗ n,n, −→ C 2n, 1) design for every n ≥ 2, where K∗ n,n is the complete bipartite directed graph and −→ C 2n is a directed Hamilton cycle in K∗ n,n.