130 research outputs found
Square Integer Heffter Arrays with Empty Cells
A Heffter array is an matrix with nonzero entries
from such that each row contains filled cells and
each column contains filled cells, every row and column sum to 0, and
no element from appears twice. Heffter arrays are useful in
embedding the complete graph on an orientable surface where the
embedding has the property that each edge borders exactly one cycle and one
cycle. Archdeacon, Boothby and Dinitz proved that these arrays can be
constructed in the case when , i.e. every cell is filled. In this paper we
concentrate on square arrays with empty cells where every row sum and every
column sum is in . We solve most of the instances of this case.Comment: 20 pages, including 2 figure
On Hardness of the Joint Crossing Number
The Joint Crossing Number problem asks for a simultaneous embedding of two
disjoint graphs into one surface such that the number of edge crossings
(between the two graphs) is minimized. It was introduced by Negami in 2001 in
connection with diagonal flips in triangulations of surfaces, and subsequently
investigated in a general form for small-genus surfaces. We prove that all of
the commonly considered variants of this problem are NP-hard already in the
orientable surface of genus 6, by a reduction from a special variant of the
anchored crossing number problem of Cabello and Mohar
Vertex-Coloring with Star-Defects
Defective coloring is a variant of traditional vertex-coloring, according to
which adjacent vertices are allowed to have the same color, as long as the
monochromatic components induced by the corresponding edges have a certain
structure. Due to its important applications, as for example in the
bipartisation of graphs, this type of coloring has been extensively studied,
mainly with respect to the size, degree, and acyclicity of the monochromatic
components.
In this paper we focus on defective colorings in which the monochromatic
components are acyclic and have small diameter, namely, they form stars. For
outerplanar graphs, we give a linear-time algorithm to decide if such a
defective coloring exists with two colors and, in the positive case, to
construct one. Also, we prove that an outerpath (i.e., an outerplanar graph
whose weak-dual is a path) always admits such a two-coloring. Finally, we
present NP-completeness results for non-planar and planar graphs of bounded
degree for the cases of two and three colors
Martin Gardner's minimum no-3-in-a-line problem
In Martin Gardner's October, 1976 Mathematical Games column in Scientific
American, he posed the following problem: "What is the smallest number of
[queens] you can put on a board of side n such that no [queen] can be added
without creating three in a row, a column, or a diagonal?" We use the
Combinatorial Nullstellensatz to prove that this number is at least n, except
in the case when n is congruent to 3 modulo 4, in which case one less may
suffice. A second, more elementary proof is also offered in the case that n is
even.Comment: 11 pages; lower bound in main theorem corrected to n-1 (from n) in
the case of n congruent to 3 mod 4, minor edits, added journal referenc
Irreducible triangulations of surfaces with boundary
A triangulation of a surface is irreducible if no edge can be contracted to
produce a triangulation of the same surface. In this paper, we investigate
irreducible triangulations of surfaces with boundary. We prove that the number
of vertices of an irreducible triangulation of a (possibly non-orientable)
surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was
known only for surfaces without boundary (b=0). While our technique yields a
worse constant in the O(.) notation, the present proof is elementary, and
simpler than the previous ones in the case of surfaces without boundary
A topological classification of convex bodies
The shape of homogeneous, generic, smooth convex bodies as described by the
Euclidean distance with nondegenerate critical points, measured from the center
of mass represents a rather restricted class M_C of Morse-Smale functions on
S^2. Here we show that even M_C exhibits the complexity known for general
Morse-Smale functions on S^2 by exhausting all combinatorial possibilities:
every 2-colored quadrangulation of the sphere is isomorphic to a suitably
represented Morse-Smale complex associated with a function in M_C (and vice
versa). We prove our claim by an inductive algorithm, starting from the path
graph P_2 and generating convex bodies corresponding to quadrangulations with
increasing number of vertices by performing each combinatorially possible
vertex splitting by a convexity-preserving local manipulation of the surface.
Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist,
this algorithm not only proves our claim but also generalizes the known
classification scheme in [36]. Our expansion algorithm is essentially the dual
procedure to the algorithm presented by Edelsbrunner et al. in [21], producing
a hierarchy of increasingly coarse Morse-Smale complexes. We point out
applications to pebble shapes.Comment: 25 pages, 10 figure
Planar Graph Coloring with Forbidden Subgraphs: Why Trees and Paths Are Dangerous
We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem.
We present a complete picture for the case with a single forbidden connected (induced or non-induced) subgraph. The 2-coloring problem is NP-hard if the forbidden subgraph is a tree with at least two edges, and it is polynomially solvable in all other cases. The 3-coloring problem is NP-hard if the forbidden subgraph is a path, and it is polynomially solvable in all other cases. We also derive results for several forbidden sets of cycles
Local chromatic number of quadrangulations of surfaces
The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four.
Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces
replacing the condition of the graph being not bipartite by a more technical condition of
an odd quadrangulation. This paper investigates when these general results are true for the
local chromatic number instead of the chromatic number. Surprisingly, we find out that
(unlike in the case of the chromatic number) this depends on the genus of the surface. For
the non-orientable surfaces of genus at most four, the local chromatic number of any odd
quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5
or higher.
We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of
arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for
the usual chromatic number
Special Feature CKD as a Model for Improving Chronic Disease Care through Electronic Health Records
Abstract Electronic health records have the potential to improve the care of patients with chronic medical conditions. CKD provides a unique opportunity to show this potential: the disease is common in the United States, there is significant room to improve CKD detection and management, CKD and its related conditions are defined primarily by objective laboratory data, CKD care requires collaboration by a diverse team of health care professionals, and improved access to CKD-related data would enable identification of a group of patients at high risk for multiple adverse outcomes. However, to realize the potential for improvement in CKD-related care, electronic health records will need to provide optimal functionality for providers and patients and interoperability across multiple health care settings. The goal of the National Kidney Disease Education Program Health Information Technology Working Group is to enable and support the widespread interoperability of data related to kidney health among health care software applications to optimize CKD detection and management. Over the course of the last 2 years, group members met to identify general strategies for using electronic health records to improve care for patients with CKD. This paper discusses these strategies and provides general goals for appropriate incorporation of CKD-related data into electronic health records and corresponding design features that may facilitate (1) optimal care of individual patients with CKD through improved access to clinical information and decision support, (2) clinical quality improvement through enhanced population management capabilities, (3) CKD surveillance to improve public health through wider availability of population-level CKD data, and (4) research to improve CKD management practices through efficiencies in study recruitment and data collection. Although these strategies may be most effectively applied in the setting of CKD, because it is primarily defined by laboratory abnormalities and therefore, an ideal computable electronic health record phenotype, they may also apply to other chronic diseases
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