We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the class...

We present tournament results and several powerful strategies for the Iterated Prisoner's Dilemma created using reinforcement learning techniques (evolutionary and particle swarm algorithms). These strategies are trained to perform well against a corpus of over 170 distinct opponents, including many well-known and classic strategies. All the trained strategies win standard tournaments against t...

The paper reveals that the Weighted Logarithmic Least Squares Method used for deriving evaluations of alternatives from incomplete pairwise comparison matrices (iWLLSM) coincides with the recursive Buchholz ranking method applied in generalized tournaments, despite their different approach and calculation. We study it with respect to a set of new properties and present the strength of the metho...

We design the first polynomial time approximation schemes (PTASs) for the Minimum Betweenness problem in tournaments and some related higher arity ranking problems. This settles the approximation status of the Betweenness problem in tournaments along with other ranking problems which were open for some time now. The results depend on a new technique of dealing with fragile ranking constraints a...

We characterize when a voting rule deened over two tournaments can be represented by sophisticated voting on a binary tree. This characterization has a particularly simple and intuitive specialization to the case that the two tournaments diier in exactly one place. These characterizations allow us to demonstrate the tree that implements a particular voting rule, and to show the minimality of an...

We prove that a regular tournament with n vertices has more than n 2 11.5 (1 − o(1)) pairwise arc-disjoint directed triangles. On the other hand, we construct regular tournaments with a feedback arc set of size less than n 2 8 , so these tournaments do not have n 8 pairwise arc-disjoint triangles. These improve upon the best known bounds for this problem.

We introduce the convex circuit-free coloration and convex circuit-free chromatic number −→ χa( −→ G) of an oriented graph −→ G and establish various basic results. We show that the problem of deciding if an oriented graph verifies χa( −→ G ) ≤ k is NP-complete if k ≥ 3 and polynomial if k ≤ 2. We exhibit an algorithm which finds the optimal convex circuit-free coloration for tournaments, and c...