The ratio of the longest cycle and longest path in semicomplete multipartite digraphs

AbstractA digraph obtained by replacing each edge of a complete n-partite graph by an arc or a pair of mutually opposite arcs is called a semicomplete n-partite digraph. We call α(D)=max1⩽i⩽n{|Vi|} the independence number of the semicomplete n-partite digraph D, where V1,V2,…,Vn are the partite sets of D. Let p and c, respectively, denote the number of vertices in a longest directed path and the number of vertices in a longest directed cycle of a digraph D. Recently, Gutin and Yeo proved that c⩾(p+1)/2 for every strongly connected semicomplete n-partite digraph D. In this paper we present for the special class of semicomplete n-partite digraphs D with connectivity κ(D)=α(D)−1⩾1 the better boundc⩾κ(D)κ(D)+1(p+1).In addition, we present examples which show that this bound is best possible

Linear Orderings of Sparse Graphs

The Linear Ordering problem consists in finding a total ordering of the vertices of a directed graph such that the number of backward arcs, i.e., arcs whose heads precede their tails in the ordering, is minimized. A minimum set of backward arcs corresponds to an optimal solution to the equivalent Feedback Arc Set problem and forms a minimum Cycle Cover. Linear Ordering and Feedback Arc Set are classic NP-hard optimization problems and have a wide range of applications. Whereas both problems have been studied intensively on dense graphs and tournaments, not much is known about their structure and properties on sparser graphs. There are also only few approximative algorithms that give performance guarantees especially for graphs with bounded vertex degree. This thesis fills this gap in multiple respects: We establish necessary conditions for a linear ordering (and thereby also for a feedback arc set) to be optimal, which provide new and fine-grained insights into the combinatorial structure of the problem. From these, we derive a framework for polynomial-time algorithms that construct linear orderings which adhere to one or more of these conditions. The analysis of the linear orderings produced by these algorithms is especially tailored to graphs with bounded vertex degrees of three and four and improves on previously known upper bounds. Furthermore, the set of necessary conditions is used to implement exact and fast algorithms for the Linear Ordering problem on sparse graphs. In an experimental evaluation, we finally show that the property-enforcing algorithms produce linear orderings that are very close to the optimum and that the exact representative delivers solutions in a timely manner also in practice. As an additional benefit, our results can be applied to the Acyclic Subgraph problem, which is the complementary problem to Feedback Arc Set, and provide insights into the dual problem of Feedback Arc Set, the Arc-Disjoint Cycles problem