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On regular vertices on the union of planar objects

2007
*
Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07
*

We establish several geometric extensions

doi:10.1145/1247069.1247110
dblp:conf/compgeom/EzraPS07
fatcat:kybchui2k5dl7gfygnr4cu5fnq
*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

2008
*
Discrete & Computational Geometry
*

We show that

doi:10.1007/s00454-008-9118-2
fatcat:plqh22mm3vbtxgtpwslcmdh4r4
*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

1972
*
13th Annual Symposium on Switching and Automata Theory (swat 1972)
*

Graphs with distinguished

doi:10.1109/swat.1972.26
dblp:conf/focs/Maibaum72
fatcat:pl3trzkti5gatlaei5ihljszoq
*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

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 [2] ), B. Fawcett (

*on*

*the*subject

*of*simple graphs), Z. ...

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

2019
*
arXiv
*
pre-print

By using known examples

arXiv:1506.06564v5
fatcat:gpljlpxywfb3zd4hk7ebwpeihy
*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]

1996
*
Lecture Notes in Computer Science
*

Acknowledgments

doi:10.1007/bfb0021790
fatcat:esad4m7oibde5m37cg6iywetp4
*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]

2014
*
Lecture Notes in Computer Science
*

We study

doi:10.1007/978-3-662-45803-7_6
fatcat:4qflfetdxre7tpsafwf3lmi2qm
*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

1999
*
Journal of Visual Communication and Image Representation
*

A description

doi:10.1006/jvci.1999.0415
fatcat:2tngmtnb7rgihhltnw4akt2fuy
*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

1999
*
Proceedings of the fifteenth annual symposium on Computational geometry - SCG '99
*

We show that

doi:10.1145/304893.304958
dblp:conf/compgeom/Efrat99
fatcat:ikexr7tsjrcrfnyxyjusvxdd6y
*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

2001
*
Discrete & Computational Geometry
*

Let R (C) denote

doi:10.1007/s00454-001-0001-7
fatcat:v2snvsgmwbcudcf6u3tyla5jka
*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]

2016
*
arXiv
*
pre-print

We exhibit an infinite family

arXiv:1604.01934v1
fatcat:b2q47o5fzzds7npw55s5iehwbq
*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

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*[4] . ...

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

2015
*
arXiv
*
pre-print

For a degree d self branched covering

arXiv:1504.01154v1
fatcat:e35r4m7km5dnjo5bklknm5vpd4
*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

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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