69 research outputs found
Stick index of knots and links in the cubic lattice
The cubic lattice stick index of a knot type is the least number of sticks
necessary to construct the knot type in the 3-dimensional cubic lattice. We
present the cubic lattice stick index of various knots and links, including all
(p,p+1)-torus knots, and show how composing and taking satellites can be used
to obtain the cubic lattice stick index for a relatively large infinite class
of knots. Additionally, we present several bounds relating cubic lattice stick
index to other known invariants.Comment: 16 pages, 12 figure
Untangling polygons and graphs
Untangling is a process in which some vertices of a planar graph are moved to
obtain a straight-line plane drawing. The aim is to move as few vertices as
possible. We present an algorithm that untangles the cycle graph C_n while
keeping at least \Omega(n^{2/3}) vertices fixed. For any graph G, we also
present an upper bound on the number of fixed vertices in the worst case. The
bound is a function of the number of vertices, maximum degree and diameter of
G. One of its consequences is the upper bound O((n log n)^{2/3}) for all
3-vertex-connected planar graphs.Comment: 11 pages, 3 figure
Obstacle Numbers of Planar Graphs
Given finitely many connected polygonal obstacles in the
plane and a set of points in general position and not in any obstacle, the
{\em visibility graph} of with obstacles is the (geometric)
graph with vertex set , where two vertices are adjacent if the straight line
segment joining them intersects no obstacle. The obstacle number of a graph
is the smallest integer such that is the visibility graph of a set of
points with obstacles. If is planar, we define the planar obstacle
number of by further requiring that the visibility graph has no crossing
edges (hence that it is a planar geometric drawing of ). In this paper, we
prove that the maximum planar obstacle number of a planar graph of order is
, the maximum being attained (in particular) by maximal bipartite planar
graphs. This displays a significant difference with the standard obstacle
number, as we prove that the obstacle number of every bipartite planar graph
(and more generally in the class PURE-2-DIR of intersection graphs of straight
line segments in two directions) of order at least is .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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
Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time
Thomassen characterized some 1-plane embedding as the forbidden configuration
such that a given 1-plane embedding of a graph is drawable in straight-lines if
and only if it does not contain the configuration [C. Thomassen, Rectilinear
drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988].
In this paper, we characterize some 1-plane embedding as the forbidden
configuration such that a given 1-plane embedding of a graph can be re-embedded
into a straight-line drawable 1-plane embedding of the same graph if and only
if it does not contain the configuration. Re-embedding of a 1-plane embedding
preserves the same set of pairs of crossing edges.
We give a linear-time algorithm for finding a straight-line drawable 1-plane
re-embedding or the forbidden configuration.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016). This is an extended
abstract. For a full version of this paper, see Hong S-H, Nagamochi H.:
Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time,
Technical Report TR 2016-002, Department of Applied Mathematics and Physics,
Kyoto University (2016
Convex drawings of graphs with non-convex boundary
Abstract. In this paper, we study a new problem of finding a convex drawing of graphs with a non-convex boundary. It is proved that every triconnected plane graph whose boundary is fixed with a star-shaped polygon admits a drawing in which every inner facial cycle is drawn as a convex polygon. Such a drawing, called an inner-convex drawing, can be obtained in linear time.
A polynomial bound for untangling geometric planar graphs
To untangle a geometric graph means to move some of the vertices so that the
resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput.
Geom., 2002] asked if every n-vertex geometric planar graph can be untangled
while keeping at least n^\epsilon vertices fixed. We answer this question in
the affirmative with \epsilon=1/4. The previous best known bound was
\Omega((\log n / \log\log n)^{1/2}). We also consider untangling geometric
trees. It is known that every n-vertex geometric tree can be untangled while
keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was
O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170
2007] by closing this gap for untangling trees. In particular, we show that for
infinitely many values of n, there is an n-vertex geometric tree that cannot be
untangled while keeping more than 3(n^{1/2}-1) vertices fixed. Moreover, we
improve the lower bound to (n/2)^{1/2}.Comment: 14 pages, 7 figure
Duality properties of indicatrices of knots
The bridge index and superbridge index of a knot are important invariants in
knot theory. We define the bridge map of a knot conformation, which is closely
related to these two invariants, and interpret it in terms of the tangent
indicatrix of the knot conformation. Using the concepts of dual and derivative
curves of spherical curves as introduced by Arnold, we show that the graph of
the bridge map is the union of the binormal indicatrix, its antipodal curve,
and some number of great circles. Similarly, we define the inflection map of a
knot conformation, interpret it in terms of the binormal indicatrix, and
express its graph in terms of the tangent indicatrix. This duality relationship
is also studied for another dual pair of curves, the normal and Darboux
indicatrices of a knot conformation. The analogous concepts are defined and
results are derived for stick knots.Comment: 22 pages, 9 figure
Integrality gaps of integer knapsack problems
We obtain optimal lower and upper bounds for the (additive) integrality
gaps of integer knapsack problems. In a randomised setting, we show that the integrality
gap of a “typical” knapsack problem is drastically smaller than the integrality
gap that occurs in a worst case scenario
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