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### On regular vertices on the union of planar objects

Esther Ezra, Janos Pach, Micha Sharir
2007 Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07
We establish several geometric extensions of the Lipton-Tarjan separator theorem for planar graphs.  ...  These results are used to obtain various properties of intersection patterns of geometric objects in the plane.  ...  The authors would also like to thank Radoś Radoičić, Benny Sudakov, Csaba Tóth, and Géza Tóth for helpful conversations on the topics of this paper.  ...

### On Regular Vertices of the Union of Planar Convex Objects

Esther Ezra, János Pach, Micha Sharir
2008 Discrete & Computational Geometry
We show that the maximum number of regular vertices (intersection points of two boundaries that intersect twice) on the boundary of the union U of C is O * (n 4/3 ), which improves earlier bounds due to  ...  The bound is nearly tight in the worst case. In this paper, a bound of  ...  Acknowledgements The authors wish to thank Boris Aronov, Alon Efrat, and an anonymous referee for their useful comments.  ...

### The characterisation of the derivation trees of context free sets of terms as regular sets

T. S. E. Maibaum
1972 13th Annual Symposium on Switching and Automata Theory (swat 1972)
Graphs with distinguished vertices (boundary vertices or sources) 1 and 2, generated from e = 1 2 and the operations // (parallel-composition) and series-composition.  ...  Application 2 : Equation system for 2-colorable series-parallel graphs We let S σ,δ be the set of series-parallel graphs that satisfy Same (σ) and Diff (δ) S σ,δ be the set of those that satisfy Same and  ...  not Diff , etc … From the equation : S = S // S ∪ S • S ∪ e we get the equation system :  ...

### Absolute planar retracts and the four color conjecture

Pavol Hell
1974 Journal of combinatorial theory. Series B (Print)
The Four Color Conjecture (4CC) is shown to be equivalent with the existence of absolute retracts in the class B of all planar graphs and also with some other conditions on the class 8.  ...  Assuming the 4CC is true, we determine ah absolute planar retracts.  ...  ACKNOWLEDGMENTS The author wishes to acknowledge useful discussions with G. Sabidussi (the director of research for  ), B. Fawcett (on the subject of simple graphs), Z.  ...

### Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs [article]

Konrad K. Dabrowski, Francois Dross, Matthew Johnson, Daniel Paulusma
2019 arXiv   pre-print
By using known examples of non-3-choosable and non-4-choosable graphs, this enables us to classify the complexity of k-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar  ...  We may also ask if for a given k-regular list assignment L of a given graph G, there exists a colouring of G that respects L. This yields the k-Regular List Colouring problem.  ...  We thank Zdeněk Dvořák and Steven Kelk for helpful comments on earlier versions of this paper.  ...

### Drawing nice projections of objects in space [chapter]

Prosenjit Bose, Pedro Ramos, Francisco Gomez, Godfried Toussaint
1996 Lecture Notes in Computer Science
Acknowledgments The authors thank Suneeta Ramaswami, Fady Habra, Ivan Balaban, Bernard Chazelle and Mike Goodrich for correspondence. The second author thanks Lola for her support.  ...  Let K be a knot with n vertices. To study the knot's combinatorial properties, knot theorists obtain a planar graph G called the diagram of K by a regular projection of K.  ...  This planar subdivision is then pre-processed for planar point location before doing binary search on the z direction.  ...

### Simultaneous Embeddability of Two Partitions [chapter]

Jan Christoph Athenstädt, Tanja Hartmann, Martin Nöllenburg
2014 Lecture Notes in Computer Science
We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks.  ...  Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to the elements in the block  ...  We thank the anonymous reviewers for helpful comments. References  ...

### Drawing Nice Projections of Objects in Space

Prosenjit Bose, Francisco Gómez, Pedro Ramos, Godfried Toussaint
1999 Journal of Visual Communication and Image Representation
A description of the set of all directions which yield regular projections can be computed in O(n 3 log n + k) time, where k is the number of intersections of a set of quadratic arcs on the direction sphere  ...  of "nice" parallel (orthographic) projections of the object.  ...  We thank David MacDonald and Rick Hoge of the Montreal Neurological Institute and McGill University for providing data and images. The second author thanks Lola for her support.  ...

### The complexity of the union of (α, β)-covered objects

Alon Efrat
1999 Proceedings of the fifteenth annual symposium on Computational geometry - SCG '99
We show that the combinatorial complexity of the union of n (α, β)covered objects of 'constant description complexity' is O(λ s+2 (n) log 2 n log log n), where s is the maximum number of intersections  ...  between the boundaries of any pair of the given objects.  ...  We wish to thank Micha Sharir for helpful discussions and many useful ideas regards the results of this paper. Thanks also to anonymous referee for many useful remarks.  ...

### Page 6937 of Mathematical Reviews Vol. , Issue 92m [page]

1992 Mathematical Reviews
A typical example occurs when there is a straight line or circular arc which is contained in the union of the boundaries of the tiles, for then one can slide all the tiles on one side using an isometry  ...  Here the ten regular tetrahedra inscribed to a regular dodecahedron are linked together at the common vertices. The author shows that this framework allows at least two types of constrained motions.  ...

### On the number of regular vertices of the union of jordan regions

B. Aronov, A. Efrat, D. Halperin, M. Sharir
2001 Discrete & Computational Geometry
Let R (C) denote the set of regular vertices on the boundary of the union of C. We present several bounds on I R (C) 1, depending on the type of the sets of C.  ...  If the boundaries of two sets in C cross exactly twice, then their intersection points are called regular vertices of the arrangement .A(C).  ...  The goal is to obtain sharp bounds on the maximal number of regular vertices that appear on 8 U. We denote the set of regular vertices on aU by R(C).  ...

### Fullerene graphs of small diameter [article]

Diego Nicodemos, Matěj Stehlík
2016 arXiv   pre-print
We exhibit an infinite family of fullerene graphs of diameter √(4n/3), where n is the number of vertices. This disproves a conjecture of Andova and Škrekovski [MATCH Commun. Math. Comput.  ...  Chem. 70 (2013) 205-220], who conjectured that every fullerene graph on n vertices has diameter at least √(5n/3)-1.  ...  The graph D r,t is best defined using the planar dual graph. Let T be the (infinite) 6-regular planar triangulation.  ...

### Generalized 2N Color Theorem

Joseph Edward Brierly, US Army TACOM Readiness ILS Function, Warren, Michigan 48397-5000
2020 Journal of Physics & Optics Sciences
The Six-color theorem applies to three dimensional space implying that a space station could be ideally designed to have six compartments adjacent to one another allowing a door from any one of the compartments  ...  The Fourcolor theorem applies to map making by the knowledge that only four colors are necessary to color a planar map.  ...  The author proved in a previous unpublished article that in general a graph with an odd number of vertices cannot be odd regular  .  ...

### Realizations of self branched coverings of the 2-sphere [article]

J. Tomasini
2015 arXiv   pre-print
For a degree d self branched covering of the 2-sphere, a notable combinatorial invariant is an integer partition of 2d -- 2, consisting of the multiplicities of the critical points.  ...  A finer invariant is the so called Hurwitz passport. The realization problem of Hurwitz passports remain largely open till today.  ...  I also thank Tan Lei for introducing the subject, and for her constant help during the preparation of this work.  ...

### The disconnection number of a graph

Helma Gladdines, Marcel van de Vel
2011 Topology and its Applications
The algorithm of the main result has been implemented to construct the classes D n for n 8, to estimate the size of D 9 , and to obtain information on certain subclasses such as non-planar graphs (n 9)  ...  The main result characterizes the members of D n+1 in terms of four possible operations on members of D n .  ...  For instance, the simple regular graphs of D 6 are 3-regular and have 8 vertices; the simple regular graphs of D 7 are either 4-regular with 5 vertices (just K 5 ) or 3-regular with 10 vertices.  ...
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