scholarly journals ALGEBRAIC CHARACTERIZATIONS OF GRAPH IMBEDDABILITY IN SURFACES AND PSEUDOSURFACES

2006 ◽  
Vol 15 (06) ◽  
pp. 681-693 ◽  
Author(s):  
LOWELL ABRAMS ◽  
DANIEL C. SLILATY
Keyword(s):  
Connected Graph ◽  
Homology Group ◽  
Dual Graph ◽  
Connected Components ◽  
Homology Groups ◽  
The Given

Given a finite connected graph G and specifications for a closed, connected pseudosurface, we characterize when G can be imbedded in a closed, connected pseudosurface with the given specifications. The specifications for the pseudosurface are: the number of face-connected components, the number of pinches, the number of crosscaps and handles, and the dimension of the first ℤ2 homology group. The characterizations are formulated in terms of the existence of a dual graph G* on the same set of edges as G which satisfies algebraic conditions inspired by homology groups and their intersection products.

scholarly journals One-relator groups and algebras related to polyhedral products

2021 ◽  
pp. 1-20
Author(s):  
Jelena Grbić ◽  
George Simmons ◽  
Marina Ilyasova ◽  
Taras Panov
Keyword(s):  
Homology Group ◽  
Homotopy Theory ◽  
Commutator Subgroup ◽  
Homology Groups ◽  
Homological Characterization ◽  
One Relator Groups ◽  
Combinatorial Condition ◽  
The Moment ◽  
Necessary And Sufficient

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

Edge (m,k)-choosability of graphs

2021 ◽  
Author(s):  
P. Soorya ◽  
K. A. Germina
Keyword(s):  
Connected Graph ◽  
Matching Number ◽  
Graph Theoretic ◽  
Maximum Value ◽  
Choice Number ◽  
The Given ◽  
Simple Connected Graph

Let [Formula: see text] be a simple, connected graph of order [Formula: see text] and size [Formula: see text] Then, [Formula: see text] is said to be edge [Formula: see text]-choosable, if there exists a collection of subsets of the edge set, [Formula: see text] of cardinality [Formula: see text] such that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are incident. This paper initiates a study on edge [Formula: see text]-choosability of certain fundamental classes of graphs and determines the maximum value of [Formula: see text] for which the given graph [Formula: see text] is edge [Formula: see text]-choosable. Also, in this paper, the relation between edge choice number and other graph theoretic parameters is discussed and we have given a conjecture on the relation between edge choice number and matching number of a graph.

The Maximum Genus of Cartesian Products of Graphs

1974 ◽  
Vol 26 (5) ◽  
pp. 1025-1035 ◽  
Author(s):  
Joseph Zaks
Keyword(s):  
Connected Graph ◽  
Connected Components ◽  
Open Disk ◽  
Maximum Genus ◽  
Cartesian Products ◽  
Euler’S Formula ◽  
Cartesian Products Of Graphs

The maximum genus γM(G) of a connected graph G has been defined in [2] as the maximum g for which there exists an embedding h : G —> S(g), where S(g) is a compact orientable 2-manifold of genus g, such that each one of the connected components of S(g) — h(G) is homeomorphic to an open disk; such an embedding is called cellular. If G is cellularly embedded in S(g), having V vertices, E edges and F faces, then by Euler's formulaV-E + F = 2-2g.

scholarly journals Homology group on manifolds and their foldings

2010 ◽  
Vol 41 (1) ◽  
pp. 31-38 ◽  
Author(s):  
M. Abu-Saleem
Keyword(s):  
Homology Group ◽  
Homology Groups ◽  
Definition Of

In this paper, we introduce the definition of the induced unfolding on the homology group. Some types of conditional foldings restricted on the elements of the homology groups are deduced. The effect of retraction on the homology group of a manifold is dicussed. The unfolding of variation curvature of manifolds on their homology group are represented. The relations between homology group of the manifold and its folding are deduced.

Characteristic Topology Method for Quadrilateral Mesh Without Self-Intersection of Dual Cycles

2002 ◽  
Author(s):  
Junichi Shinoda ◽  
Olga Egorova ◽  
Haozhi Qu ◽  
Ichiro Hagiwara
Keyword(s):  
Initial Data ◽  
Mesh Generation ◽  
Dual Graph ◽  
Surface Mesh ◽  
Hexahedral Mesh ◽  
Quadrilateral Mesh ◽  
Hexahedral Mesh Generation ◽  
Dual Cycle ◽  
The Given ◽  
Polygonal Model

The Dual Cycle Elimination method was proposed by Mu¨ller-Hannemann for hexahedral mesh generation. The method begins with a surface quadrilateral mesh whose dual cycles have no self-intersections and, after the elimination of dual cycles, a hexahedral mesh is generated while tracing back the reverse order of eliminations and supplementing hexahedrons inside the object step by step. This paper presents the Characteristic Topology Method as a means to prescribe a quadrilateral surface mesh that can be initial data for further hexahedral mesh generation. The goal of this method is to stress the topology of the given surface and thus use construction of the loops within the algorithm. The surface is given in a nodal polygonal model and then decomposed into a triangle-quadrilateral model. Templates are used to determine the loops. Then due to some rules every loop is implemented by special additional Dual Cycles. The total mesh is the dual graph to the graph of dual cycles. The problem of self-intersections that may appear comes from Mu¨ller-Hannemann’s approach stated above and that is also implemented in this work as a sketch.

Abelian Groups with a Vanishing Homology Group

1969 ◽  
Vol 21 ◽  
pp. 406-409 ◽  
Author(s):  
James A. Schafer
Keyword(s):  
Homology Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Rational Numbers ◽  
Complete Characterization ◽  
Integral Homology ◽  
Positive Dimension ◽  
Homology Groups

In this paper, we wish to characterize those abelian groups whose integral homology groups vanish in some positive dimension. We obtain a complete characterization provided the dimension in which the homology vanishes is odd; in fact, we prove that the only abelian groups which possess a vanishing homology group in an odd dimension are, up to isomorphism, subgroups of Qn, where Q denotes the additive group of rational numbers. The case of vanishing in an even dimension is much more complicated. We exhibit a class of groups whose homology vanishes in even dimensions and is otherwise very nice, namely the subgroups of Q/Z, and then show that unless we impose further restrictions, there exist abelian groups which possess the homology of subgroups of Q/Z without being isomorphic to a subgroup of Q/Z.

A RUN-BASED ONE-AND-A-HALF-SCAN CONNECTED-COMPONENT LABELING ALGORITHM

2010 ◽  
Vol 24 (04) ◽  
pp. 557-579 ◽  
Author(s):  
LIFENG HE ◽  
YUYAN CHAO ◽  
KENJI SUZUKI
Keyword(s):  
Binary Image ◽  
Experimental Results ◽  
Connected Components ◽  
Connected Component ◽  
Run Length ◽  
Highly Efficient ◽  
Connected Component Labeling ◽  
Labeling Algorithm ◽  
The Given

This paper presents a run- and label-equivalence-based one-and-a-half-scan algorithm for labeling connected components in a binary image. Major differences between our algorithm and conventional label-equivalence-based algorithms are: (1) all conventional label-equivalence-based algorithms scan all pixels in the given image at least twice, whereas our algorithm scans background pixels once and object pixels twice; (2) all conventional label-equivalence-based algorithms assign a provisional label to each object pixel in the first scan and relabel the pixel in the later scan(s), whereas our algorithm assigns a provisional label to each run in the first scan, and after resolving label equivalences between runs, by using the recorded run data, it assigns each object pixel a final label directly. That is, in our algorithm, relabeling of object pixels is not necessary any more. Experimental results demonstrated that our algorithm is highly efficient on images with many long runs and/or a small number of object pixels. Moreover, our algorithm is directly applicable to run-length-encoded images, and we can obtain contours of connected components efficiently.

Homology Invariants

1978 ◽  
Vol 30 (03) ◽  
pp. 655-670 ◽  
Author(s):  
Richard Hartley ◽  
Kunio Murasugi
Keyword(s):  
Homology Group ◽  
Covering Space ◽  
Simple Relationship ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Homology Groups ◽  
Branched Covering ◽  
The Relationship

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.

ON SIMULTANEOUS CONSTRUCTION OF VORONOI DIAGRAM AND DELAUNAY TRIANGULATION BY PHYSARUM POLYCEPHALUM

2009 ◽  
Vol 19 (09) ◽  
pp. 3109-3117 ◽  
Author(s):  
TOMOHIRO SHIRAKAWA ◽  
ANDREW ADAMATZKY ◽  
YUKIO-PEGIO GUNJI ◽  
YOSHIHIRO MIYAKE
Keyword(s):  
Voronoi Diagram ◽  
Delaunay Triangulation ◽  
Physarum Polycephalum ◽  
Vegetative State ◽  
Dual Graph ◽  
Data Sets ◽  
Limited Data ◽  
Data Set ◽  
The Given

We experimentally demonstrate that both Voronoi diagram and its dual graph Delaunay triangulation are simultaneously constructed — for specific conditions — in cultures of plasmodium, a vegetative state of Physarum polycephalum. Every point of a given planar data set is represented by a tiny mass of plasmodium. The plasmodia spread from their initial locations but, in certain conditions, stop spreading when they encounter plasmodia originated from different locations. Thus space loci not occupied by the plasmodia represent edges of Voronoi diagram of the given planar set. At the same time, the plasmodia originating at neighboring locations form merging protoplasmic tubes, where the strongest tubes approximate Delaunay triangulation of the given planar set. The problems are solved by plasmodium only for limited data sets, however the results presented lay a sound ground for further investigations.

Eco-Architecture Analysis as a Method of End-of-Life Decision Making for Sustainable Product Design

2007 ◽  
Author(s):  
Min Jung Kwak ◽  
Yoo Suk Hong ◽  
Nam Wook Cho
Keyword(s):  
Product Design ◽  
Design Stage ◽  
Connected Components ◽  
Sustainable Product Design ◽  
Early Design Stage ◽  
Effective Manner ◽  
Sustainable Product ◽  
Novel Concept ◽  
End Of Life Decision ◽  
The Given

In recent years, sustainable product design has become a great concern to product manufacturers. An effective way to enhance the product sustainability is to design products that are easy to disassemble and recycle. An EOL strategy is concerned with how to disassemble a product and what to do with each of the resulting disassembled parts. A sound understanding of the EOL strategy from the early design stage could improve the ease of disassembly and recycling in an efficient and effective manner. We introduce a novel concept of eco-architecture which represents a scheme by which the physical components are allocated to EOL modules. An EOL module is a physical chunk of connected components or a feasible subassembly which can be simultaneously processed by the same EOL option without further disassembly. In this paper, a method for analyzing and optimizing the eco-architecture of a product in the architecture design stage is proposed. Using mathematical programming, it produces an optimal eco-architecture based on the estimation of the economic values and costs for possible EOL modules under the given environmental regulations.

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