3,573 research outputs found
Excluding pairs of tournaments
The Erd\H{o}s-Hajnal conjecture states that for every given undirected graph
there exists a constant such that every graph that does not
contain as an induced subgraph contains a clique or a stable set of size at
least . The conjecture is still open. Its equivalent directed
version states that for every given tournament there exists a constant
such that every -free tournament contains a transitive
subtournament of order at least . We prove in this paper that
-free tournaments contain transitive subtournaments of
size at least for some and several
pairs of tournaments: , . In particular we prove that
-freeness implies existence of the polynomial-size transitive
subtournaments for several tournaments for which the conjecture is still
open ( stands for the \textit{complement of }). To the best of our
knowledge these are first nontrivial results of this type
Hereditary properties of combinatorial structures: posets and oriented graphs
A hereditary property of combinatorial structures is a collection of
structures (e.g. graphs, posets) which is closed under isomorphism, closed
under taking induced substructures (e.g. induced subgraphs), and contains
arbitrarily large structures. Given a property P, we write P_n for the
collection of distinct (i.e., non-isomorphic) structures in a property P with n
vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of
P. Also, we write P^n for the collection of distinct labelled structures in P
with vertices labelled 1,...,n, and call the function n -> |P^n| the labelled
speed of P.
The possible labelled speeds of a hereditary property of graphs have been
extensively studied, and the aim of this paper is to investigate the possible
speeds of other combinatorial structures, namely posets and oriented graphs.
More precisely, we show that (for sufficiently large n), the labelled speed of
a hereditary property of posets is either 1, or exactly a polynomial, or at
least 2^n - 1. We also show that there is an initial jump in the possible
unlabelled speeds of hereditary properties of posets, tournaments and directed
graphs, from bounded to linear speed, and give a sharp lower bound on the
possible linear speeds in each case.Comment: 26 pgs, no figure
Who Can Win a Single-Elimination Tournament?
A single-elimination (SE) tournament is a popular way to select a winner in
both sports competitions and in elections. A natural and well-studied question
is the tournament fixing problem (TFP): given the set of all pairwise match
outcomes, can a tournament organizer rig an SE tournament by adjusting the
initial seeding so that their favorite player wins? We prove new sufficient
conditions on the pairwise match outcome information and the favorite player,
under which there is guaranteed to be a seeding where the player wins the
tournament. Our results greatly generalize previous results. We also
investigate the relationship between the set of players that can win an SE
tournament under some seeding (so called SE winners) and other traditional
tournament solutions. In addition, we generalize and strengthen prior work on
probabilistic models for generating tournaments. For instance, we show that
\emph{every} player in an player tournament generated by the Condorcet
Random Model will be an SE winner even when the noise is as small as possible,
; prior work only had such results for . We also establish new results for significantly more
general generative models.Comment: A preliminary version appeared in Proceedings of the 30th AAAI
Conference on Artificial Intelligence (AAAI), 201
Algorithmic Aspects of a General Modular Decomposition Theory
A new general decomposition theory inspired from modular graph decomposition
is presented. This helps unifying modular decomposition on different
structures, including (but not restricted to) graphs. Moreover, even in the
case of graphs, the terminology ``module'' not only captures the classical
graph modules but also allows to handle 2-connected components, star-cutsets,
and other vertex subsets. The main result is that most of the nice algorithmic
tools developed for modular decomposition of graphs still apply efficiently on
our generalisation of modules. Besides, when an essential axiom is satisfied,
almost all the important properties can be retrieved. For this case, an
algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is
generalised and yields a very efficient solution to the associated
decomposition problem
Incentives and Superstars on the LPGA Tour
Following Ehrenberg and Bognanno (1990a, b), this paper explores the role of incentives on the 2000 LPGA Tour. Overall, it finds them to have limited effectiveness. Several possible explanations are considered, including unmeasured differences in both abilities and courses and variations in the distribution of prizes across tournaments. The existence of a “superstar effect” is also considered.
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