Topological correlations in cellular structures and planar graph theory

1992 ◽  
Vol 69 (18) ◽  
pp. 2674-2677 ◽  
Author(s):  
C. Godrèche ◽  
I. Kostov ◽  
I. Yekutieli
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Cellular Structures ◽  
Topological Correlations

Effective coloration

1976 ◽  
Vol 41 (2) ◽  
pp. 469-480 ◽  
Author(s):  
Dwight R. Bean
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Recursive Function ◽  
Function Theory ◽  
Yield Information ◽  
Countably Infinite ◽  
Degrees Of Unsolvability ◽  
Infinite Planar

AbstractWe are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(x(p) − 1)-coloring, where x(p) is the least number of colors which will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs.

A Characterization of Almost-Planar Graphs

1996 ◽  
Vol 5 (3) ◽  
pp. 227-245 ◽  
Author(s):  
Bradley S. Gubser
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Planar Graphs ◽  
Explicit Description ◽  
Famous Result ◽  
Minor Criterion ◽  
Excluded Minor

Kuratowski's Theorem, perhaps the most famous result in graph theory, states that K5 and K3,3 are the only non-planar graphs for which both G\e, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the almost-planar graphs, those non-planar graphs for which G\e or G/e is planar for all edges e of G. This paper gives two characterizations of the almost-planar graphs: an explicit description of the structure of almost-planar graphs; and an excluded minor criterion. We also give a best possible bound on the number of edges of an almost-planar graph.

scholarly journals Extremal $H$-Free Planar Graphs

10.37236/8255 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Yongxin Lan ◽  
Yongtang Shi ◽  
Zi-Xia Song
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Sufficient Conditions ◽  
Subcubic Graph ◽  
Stone Theorem

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory  83 (2016) 213–230], that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield  $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem.  We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to  $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

scholarly journals A Survey of Stack-Sorting Disciplines

10.37236/1693 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Miklós Bóna
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Young Tableau ◽  
Simplicial Complexes ◽  
Permutation Patterns ◽  
Stack Sorting

We review the various ways that stacks, their variations and their combinations, have been used as sorting devices. In particular, we show that they have been a key motivator for the study of permutation patterns. We also show that they have connections to other areas in combinatorics such as Young tableau, planar graph theory, and simplicial complexes.

scholarly journals A note on planar Ramsey numbers for a triangle versus wheels

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Guofei Zhou ◽  
Yaojun Chen ◽  
Zhengke Miao ◽  
Shariefuddin Pirzada
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
International Audience

Graph Theory International audience For two given graphs G and H , the planar Ramsey number P R ( G; H ) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G , or its complement contains a copy of H . In this paper, we determine all planar Ramsey numbers for a triangle versus wheels.

Preparatory Procedures for Electron Microscopic Analysis at the Molecular Level of Resolution

1978 ◽  
Vol 36 (3) ◽  
pp. 516-520
Author(s):  
F.J. Sjostrand
Keyword(s):  
Electron Microscope ◽  
Molecular Level ◽  
Microscopic Analysis ◽  
Resolving Power ◽  
Cellular Structures ◽  
Electron Microscopic ◽  
Systematic Analysis ◽  
Technical Developments ◽  
Thin Sectioning ◽  
Obvious Reason

In the 1940's and 1950's electron microscopy conferences were attended with everybody interested in learning about the latest technical developments for one very obvious reason. There was the electron microscope with its outstanding performance but nobody could make very much use of it because we were lacking proper techniques to prepare biological specimens. The development of the thin sectioning technique with its perfectioning in 1952 changed the situation and systematic analysis of the structure of cells could now be pursued. Since then electron microscopists have in general become satisfied with the level of resolution at which cellular structures can be analyzed when applying this technique. There has been little interest in trying to push the limit of resolution closer to that determined by the resolving power of the electron microscope.

Some aspects of the defect structure in an austenitic stainless steel

1978 ◽  
Vol 36 (1) ◽  
pp. 314-315
Author(s):  
R. Gonzalez ◽  
L. Bru
Keyword(s):  
Stainless Steel ◽  
Austenitic Stainless Steel ◽  
Cellular Structures ◽  
Total Strain Amplitude ◽  
Total Deformation ◽  
Density Of Dislocations ◽  
Planar Arrays ◽  
Stacking Fault Tetrahedra ◽  
Distribution Of Dislocations ◽  
Very High

The analysis of stacking fault tetrahedra (SFT) in fatigued metals (1,2) is somewhat complicated, due partly to their relatively low density, but principally to the presence of a very high density of dislocations which hides them. In order to overcome this second difficulty, we have used in this work an austenitic stainless steel that deforms in a planar mode and, as expected, examination of the substructure revealed planar arrays of dislocation dipoles rather than the cellular structures which appear both in single and polycrystals of cyclically deformed copper and silver. This more uniform distribution of dislocations allows a better identification of the SFT.The samples were fatigue deformed at the constant total strain amplitude Δε = 0.025 for 5 cycles at three temperatures: 85, 293 and 773 K. One of the samples was tensile strained with a total deformation of 3.5%.

Computational Micrograph Registration with Sieve Processes

1996 ◽  
Vol 54 ◽  
pp. 440-441
Author(s):  
P.J. Phillips ◽  
J. Huang ◽  
S. M. Dunn
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Spatial Organization ◽  
Spatial Arrangement ◽  
Stereo Image ◽  
Image Model ◽  
Geometric Invariants ◽  
Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

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