232 research outputs found
Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings)
Given an embedded planar acyclic digraph G, we define the problem of "acyclic
hamiltonian path completion with crossing minimization (Acyclic-HPCCM)" to be
the problem of determining an hamiltonian path completion set of edges such
that, when these edges are embedded on G, they create the smallest possible
number of edge crossings and turn G to a hamiltonian digraph. Our results
include:
--We provide a characterization under which a triangulated st-digraph G is
hamiltonian.
--For an outerplanar triangulated st-digraph G, we define the st-polygon
decomposition of G and, based on its properties, we develop a linear-time
algorithm that solves the Acyclic-HPCCM problem with at most one crossing per
edge of G.
--For the class of st-planar digraphs, we establish an equivalence between
the Acyclic-HPCCM problem and the problem of determining an upward 2-page
topological book embedding with minimum number of spine crossings. We infer
(based on this equivalence) for the class of outerplanar triangulated
st-digraphs an upward topological 2-page book embedding with minimum number of
spine crossings and at most one spine crossing per edge.
To the best of our knowledge, it is the first time that edge-crossing
minimization is studied in conjunction with the acyclic hamiltonian completion
problem and the first time that an optimal algorithm with respect to spine
crossing minimization is presented for upward topological book embeddings
Upward Book Embeddings of st-Graphs
We study k-page upward book embeddings (kUBEs) of st-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on k pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a kUBE is NP-complete for k >= 3. A hardness result for this problem was previously known only for k = 6 [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on k=2. On the algorithmic side, we present polynomial-time algorithms for testing the existence of 2UBEs of planar st-graphs with branchwidth b and of plane st-graphs whose faces have a special structure. These algorithms run in O(f(b)* n+n^3) time and O(n) time, respectively, where f is a singly-exponential function on b. Moreover, on the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving 2UBE
Crossing-Free Acyclic Hamiltonian Path Completion for Planar st-Digraphs
In this paper we study the problem of existence of a crossing-free acyclic
hamiltonian path completion (for short, HP-completion) set for embedded upward
planar digraphs. In the context of book embeddings, this question becomes:
given an embedded upward planar digraph , determine whether there exists an
upward 2-page book embedding of preserving the given planar embedding.
Given an embedded -digraph which has a crossing-free HP-completion
set, we show that there always exists a crossing-free HP-completion set with at
most two edges per face of . For an embedded -free upward planar digraph
, we show that there always exists a crossing-free acyclic HP-completion set
for which, moreover, can be computed in linear time. For a width-
embedded planar -digraph , we show that we can be efficiently test
whether admits a crossing-free acyclic HP-completion set.Comment: Accepted to ISAAC200
Continuity argument revisited: geometry of root clustering via symmetric products
We study the spaces of polynomials stratified into the sets of polynomial
with fixed number of roots inside certain semialgebraic region , on its
border, and at the complement to its closure. Presented approach is a
generalisation, unification and development of several classical approaches to
stability problems in control theory: root clustering (-stability) developed
by R.E. Kalman, B.R. Barmish, S. Gutman et al., -decomposition(Yu.I.
Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A.
Fam, J. Meditch, J.Ackermann).
Our approach is based on the interpretation of correspondence between roots
and coefficients of a polynomial as a symmetric product morphism.
We describe the topology of strata up to homotopy equivalence and, for many
important cases, up to homeomorphism. Adjacencies between strata are also
described. Moreover, we provide an explanation for the special position of
classical stability problems: Hurwitz stability, Schur stability,
hyperbolicity.Comment: 45 pages, 4 figure
On 1-bend Upward Point-set Embeddings of -digraphs
We study the upward point-set embeddability of digraphs on one-sided convex
point sets with at most 1 bend per edge. We provide an algorithm to compute a
1-bend upward point-set embedding of outerplanar -digraphs on arbitrary
one-sided convex point sets. We complement this result by proving that for
every there exists a -outerplanar -digraph with
vertices and a one-sided convex point set so that does not admit a
1-bend upward point-set embedding on
Computing upward topological book embeddings of upward planar digraphs
This paper studies the problem of computing an upward topological book embedding of an upward planar digraph G, i.e. a topological book embedding of G where all edges are monotonically increasing in the upward direction. Besides having its own inherent interest in the theory of upward book embeddability, the question has applications to well studied research topics of computational geometry and of graph drawing. The main results of the paper are as follows. -Every upward planar digraph G with n vertices admits an upward topological book embedding such that every edge of G crosses the spine of the book at most once. -Every upward planar digraph G with n vertices admits a point-set embedding on any set of n distinct points in the plane such that the drawing is upward and every edge of G has at most two bends. -Every pair of upward planar digraphs sharing the same set of n vertices admits an upward simultaneous embedding with at most two bends per edge
Developing a Mathematical Model for Bobbin Lace
Bobbin lace is a fibre art form in which intricate and delicate patterns are
created by braiding together many threads. An overview of how bobbin lace is
made is presented and illustrated with a simple, traditional bookmark design.
Research on the topology of textiles and braid theory form a base for the
current work and is briefly summarized. We define a new mathematical model that
supports the enumeration and generation of bobbin lace patterns using an
intelligent combinatorial search. Results of this new approach are presented
and, by comparison to existing bobbin lace patterns, it is demonstrated that
this model reveals new patterns that have never been seen before. Finally, we
apply our new patterns to an original bookmark design and propose future areas
for exploration.Comment: 20 pages, 18 figures, intended audience includes Artists as well as
Computer Scientists and Mathematician
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