3,576 research outputs found
Games on interval and permutation graph representations
We describe combinatorial games on graphs in which two players antagonistically build a representation of a subgraph of a given graph. We show that for a large class of these games, determining whether a given instance is a winning position for the next player is PSPACE-hard. In contrast, we give polynomial time algorithms for solving some versions of the games on trees
Equivalences on Acyclic Orientations
The cyclic and dihedral groups can be made to act on the set Acyc(Y) of
acyclic orientations of an undirected graph Y, and this gives rise to the
equivalence relations ~kappa and ~delta, respectively. These two actions and
their corresponding equivalence classes are closely related to combinatorial
problems arising in the context of Coxeter groups, sequential dynamical
systems, the chip-firing game, and representations of quivers.
In this paper we construct the graphs C(Y) and D(Y) with vertex sets Acyc(Y)
and whose connected components encode the equivalence classes. The number of
connected components of these graphs are denoted kappa(Y) and delta(Y),
respectively. We characterize the structure of C(Y) and D(Y), show how delta(Y)
can be derived from kappa(Y), and give enumeration results for kappa(Y).
Moreover, we show how to associate a poset structure to each kappa-equivalence
class, and we characterize these posets. This allows us to create a bijection
from Acyc(Y)/~kappa to the union of Acyc(Y')/~kappa and Acyc(Y'')/~kappa, Y'
and Y'' denote edge deletion and edge contraction for a cycle-edge in Y,
respectively, which in turn shows that kappa(Y) may be obtained by an
evaluation of the Tutte polynomial at (1,0).Comment: The original paper was extended, reorganized, and split into two
papers (see also arXiv:0802.4412
Efficient computation of the Shapley value for game-theoretic network centrality
The Shapley value—probably the most important normative payoff division scheme in coalitional games—has recently been advocated as a useful measure of centrality in networks. However, although this approach has a variety of real-world applications (including social and organisational networks, biological networks and communication networks), its computational properties have not been widely studied. To date, the only practicable approach to compute Shapley value-based centrality has been via Monte Carlo simulations which are computationally expensive and not guaranteed to give an exact answer. Against this background, this paper presents the first study of the computational aspects of the Shapley value for network centralities. Specifically, we develop exact analytical formulae for Shapley value-based centrality in both weighted and unweighted networks and develop efficient (polynomial time) and exact algorithms based on them. We empirically evaluate these algorithms on two real-life examples (an infrastructure network representing the topology of the Western States Power Grid and a collaboration network from the field of astrophysics) and demonstrate that they deliver significant speedups over the Monte Carlo approach. Fo
Inflations of Geometric Grid Classes: Three Case Studies
We enumerate three specific permutation classes defined by two forbidden
patterns of length four. The techniques involve inflations of geometric grid
classes
Ziggurats and Rotation Numbers
We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's theorem (i.e., the conjecture of Jankins-Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces
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