Minimal Stable Sets in Tournaments

We propose a systematic methodology for defining tournament solutions as extensions of maximality. The central concepts of this methodology are maximal qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy of tournament solutions, which encompasses the top cycle, the uncovered set, the Banks set, the minimal covering set, the tournament equilibrium set, the Copeland set, and the bipartisan set. Moreover, the hierarchy includes a new tournament solution, the minimal extending set, which is conjectured to refine both the minimal covering set and the Banks set.Comment: 29 pages, 4 figures, changed conten

Hereditary properties of tournaments

A collection of unlabelled tournaments P is called a hereditary property if it is closed under isomorphism and under taking induced sub-tournaments. The speed of P is the function n -> |P_n|, where P_n = {T \in P : |V(T)| = n}. In this paper, we prove that there is a jump in the possible speeds of a hereditary property of tournaments, from polynomial to exponential speed. Moreover, we determine the minimal exponential speed, |P_n| = c^(n + o(n)), where c = 1.47... is the largest real root of the polynomial x^3 = x^2 + 1, and the unique hereditary property with this speed.Comment: 28 pgs, 2 figures, submitted November 200

Reconstruction of complete interval tournaments

Let $a, b$ and $n$ be nonnegative integers $(b \geq a, \ b > 0, \ n \geq 1)$, $\mathcal{G}_n(a,b)$ be a multigraph on $n$ vertices in which any pair of vertices is connected with at least $a$ and at most $b$ edges and \textbf{v =} $(v_1, v_2, ..., v_n)$ be a vector containing $n$ nonnegative integers. We give a necessary and sufficient condition for the existence of such orientation of the edges of $\mathcal{G}_n(a,b)$, that the resulted out-degree vector equals to \textbf{v}. We describe a reconstruction algorithm. In worst case checking of \textbf{v} requires $\Theta(n)$ time and the reconstruction algorithm works in $O(bn^3)$ time. Theorems of H. G. Landau (1953) and J. W. Moon (1963) on the score sequences of tournaments are special cases $b = a = 1$ resp. $b = a \geq 1$ of our result