182 research outputs found
Local Restrictions for Various Classes of Directed Graphs
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135364/1/jlms0087.pd
Optimal strategies for a game on amenable semigroups
The semigroup game is a two-person zero-sum game defined on a semigroup S as
follows: Players 1 and 2 choose elements x and y in S, respectively, and player
1 receives a payoff f(xy) defined by a function f from S to [-1,1]. If the
semigroup is amenable in the sense of Day and von Neumann, one can extend the
set of classical strategies, namely countably additive probability measures on
S, to include some finitely additive measures in a natural way. This extended
game has a value and the players have optimal strategies. This theorem extends
previous results for the multiplication game on a compact group or on the
positive integers with a specific payoff. We also prove that the procedure of
extending the set of allowed strategies preserves classical solutions: if a
semigroup game has a classical solution, this solution solves also the extended
game.Comment: 17 pages. To appear in International Journal of Game Theor
Quantum entanglement between a nonlinear nanomechanical resonator and a microwave field
We consider a theoretical model for a nonlinear nanomechanical resonator
coupled to a superconducting microwave resonator. The nanomechanical resonator
is driven parametrically at twice its resonance frequency, while the
superconducting microwave resonator is driven with two tones that differ in
frequency by an amount equal to the parametric driving frequency. We show that
the semi-classical approximation of this system has an interesting fixed point
bifurcation structure. In the semi-classical dynamics a transition from stable
fixed points to limit cycles is observed as one moves from positive to negative
detuning. We show that signatures of this bifurcation structure are also
present in the full dissipative quantum system and further show that it leads
to mixed state entanglement between the nanomechanical resonator and the
microwave cavity in the dissipative quantum system that is a maximum close to
the semi-classical bifurcation. Quantum signatures of the semi-classical
limit-cycles are presented.Comment: 36 pages, 18 figure
Graph states in phase space
The phase space for a system of qubits is a discrete grid of points, whose axes are labeled in terms of the elements of the
finite field \Gal{2^n} to endow it with proper geometrical properties. We
analyze the representation of graph states in that phase space, showing that
these states can be identified with a class of non-singular curves. We provide
an algebraic representation of the most relevant quantum operations acting on
these states and discuss the advantages of this approach.Comment: 14 pages. 2 figures. Published in Journal of Physics
INDEPENDENT DISCOVERIES IN GRAPH THEORY *
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/72829/1/j.1749-6632.1979.tb17761.x.pd
Polynomial kernelization for removing induced claws and diamonds
A graph is called (claw,diamond)-free if it contains neither a claw (a
) nor a diamond (a with an edge removed) as an induced subgraph.
Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of
triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex
is in at most two maximal cliques and every edge is in exactly one maximal
clique.
In this paper we consider the parameterized complexity of the
(claw,diamond)-free Edge Deletion problem, where given a graph and a
parameter , the question is whether one can remove at most edges from
to obtain a (claw,diamond)-free graph. Our main result is that this problem
admits a polynomial kernel. We complement this finding by proving that, even on
instances with maximum degree , the problem is NP-complete and cannot be
solved in time unless the Exponential Time
Hypothesis fai
The Complexity of the Empire Colouring Problem
We investigate the computational complexity of the empire colouring problem
(as defined by Percy Heawood in 1890) for maps containing empires formed by
exactly countries each. We prove that the problem can be solved in
polynomial time using colours on maps whose underlying adjacency graph has
no induced subgraph of average degree larger than . However, if , the problem is NP-hard even if the graph is a forest of paths of arbitrary
lengths (for any , provided .
Furthermore we obtain a complete characterization of the problem's complexity
for the case when the input graph is a tree, whereas our result for arbitrary
planar graphs fall just short of a similar dichotomy. Specifically, we prove
that the empire colouring problem is NP-hard for trees, for any , if
(and polynomial time solvable otherwise). For arbitrary
planar graphs we prove NP-hardness if for , and , for . The result for planar graphs also proves the NP-hardness of colouring
with less than 7 colours graphs of thickness two and less than colours
graphs of thickness .Comment: 23 pages, 12 figure
Emergence of Symmetry in Complex Networks
Many real networks have been found to have a rich degree of symmetry, which
is a very important structural property of complex network, yet has been rarely
studied so far. And where does symmetry comes from has not been explained. To
explore the mechanism underlying symmetry of the networks, we studied
statistics of certain local symmetric motifs, such as symmetric bicliques and
generalized symmetric bicliques, which contribute to local symmetry of
networks. We found that symmetry of complex networks is a consequence of
similar linkage pattern, which means that nodes with similar degree tend to
share similar linkage targets. A improved version of BA model integrating
similar linkage pattern successfully reproduces the symmetry of real networks,
indicating that similar linkage pattern is the underlying ingredient that
responsible for the emergence of the symmetry in complex networks.Comment: 7 pages, 7 figure
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