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Connecting colored point sets

2007
*
Discrete Applied Mathematics
*

Let S = B ∪ R be a two-

doi:10.1016/j.dam.2006.06.010
fatcat:q6d7trg6nrff5lj3yzh7mb5ju4
*colored**set*of n*points*in the plane. ... This improves over a previous method in Tokunaga [Intersection number of two*connected*geometric graphs, Inform. Process. ... Acknowledgments Thanks go to Mark van Krefeld for mentioning to us the problem of generating*colored*spanning trees by flipping, and to Ferran Hurtado for*pointing*us to Ref. [16] . ...##
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Quadrangulations and 2-colorations

2005
*
European Workshop on Computational Geometry
*

Any metric quadrangulation (made by segments of straight line) of a

dblp:conf/ewcg/CortesMNV05
fatcat:luqzdpvrvzc43f752sy3ju2ri4
*point**set*in the plane determines a 2-*coloration*of the*set*, such that edges of the quadrangulation can only join*points*with different ... Although the answer is negative in general, we can show a very wide family of 2-*colorations*, called onions 2-*coloration*, that are quadrangulable and which graph of quadrangulations is always*connected*. ...*point**set*is, in general, not*connected*. ...##
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Colored Point-set Embeddings of Acyclic Graphs
[article]

2017
*
arXiv
*
pre-print

The lower bound holds even when the function that maps vertices to

arXiv:1708.09167v1
fatcat:sudm3oikyzabpkwykcfiymxzru
*points*is not a bijection but it is defined by a 3-*coloring*. ... In contrast, a constant number of bends per edge can be obtained for 3-*colored*paths and for 3-*colored*caterpillars whose leaves all have the same*color*. ... In this case we*connect*the roots of the stars and obtain a 2-*colored*caterpillar, which admits a 2-*colored**point*-*set*embedding on any 2-*colored**set*of*points*with curve complexity two [4] . ...##
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Colored Point-Set Embeddings of Acyclic Graphs
[chapter]

2018
*
Lecture Notes in Computer Science
*

The lower bound holds even when the function that maps vertices to

doi:10.1007/978-3-319-73915-1_32
fatcat:fbv2cmuppjcjjd5czn2n2dvn5i
*points*is not a bijection but it is defined by a 3-*coloring*. ... In contrast, a constant number of bends per edge can be obtained for 3-*colored*paths and for 3-*colored*caterpillars whose leaves all have the same*color*. ... In this case we*connect*the roots of the stars and obtain a 2-*colored*caterpillar, which admits a 2-*colored**point*-*set*embedding on any 2-*colored**set*of*points*with curve complexity two [4] . ...##
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Combinatorial Species and Cluster Expansions

2010
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Moscow Mathematical Journal
*

This paper will survey recent progress on clarifying the

doi:10.17323/1609-4514-2010-10-4-713-727
fatcat:bjjkcpd23zcunpkkqjnh22n5pa
*connection*between enumerative combinatorics and cluster expansions. ... One-*point**set*of*color*p indicator species X p . ... The one-*point**set*of*color*p indicator species X p on the*colored**set*a : U → P has the value ∅ unless U consists of a one-*point**set*U = {j} with a(j) = p. ...##
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On the enumeration of finite maximal connected topologies

1973
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Journal of combinatorial theory. Series B (Print)
*

In this paper, it is shown that the number of homeomorphism classes of maximal

doi:10.1016/0095-8956(73)90020-8
fatcat:vnp6ibdcxnecjap4oms2bsdz5m
*connected*topologies defined on a*set*with n*points*is equal to twice the number of n*point*trees minus the number of n*point*... Another result is that a chain of*connected*n*point*T" topologies, linearly ordered by strict fineness, can contain a maximum of $(n" -3n + 4) topologies, and, moreover, this number is the best possible ... by either (A)*connecting*a*point*01*colored*1 to a suitable*point*/3 with*color*2 in G(e) or (B)*connecting*a*point*a:*colored*2 to a suitable*point*/3 with*color*1 in G(e). ...##
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Combinatorics and cluster expansions

2010
*
Probability Surveys
*

This article is about the

doi:10.1214/10-ps159
fatcat:f2yqv7dxjbcepjk3siceiv6lxa
*connection*between enumerative combinatorics and equilibrium statistical mechanics. ... It turns out that other problems in combinatorics and statistical mechanics can be translated to this gas*setting*, so it is a universal prescription for dealing with systems of high dimension. ... The one-*point**colored**set*of designated*color*indicator species X p The one-*point**set*indicator species X p assigns to each*colored**set*a : U → P the empty*set*of structures, with the exception of the ...##
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On Embeddability of Buses in Point Sets
[article]

2015
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arXiv
*
pre-print

*Set*membership of

*points*in the plane can be visualized by

*connecting*corresponding

*points*via graphical features, like paths, trees, polygons, ellipses. ... , such that all

*points*with the same

*color*are

*connected*with vertical line segments to their bus. ... We assume the

*sets*to be given by single-

*colored*

*points*, such that in the final visualization, called bus realization, every

*point*of the same

*color*is

*connected*to exactly one bus associated with this ...

##
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On regulating sets and the disparity of planar cubic graphs

1974
*
Canadian mathematical bulletin
*

Let G be a

doi:10.4153/cmb-1974-067-2
fatcat:ukwxq4cn2ja4tgyify3vcv4w5u
*connected*graph with m*points*and let V 0 ={v i9 w t | i= 1,... ,n} be a*set*of 2n arbitrarily chosen distinct*points*of G, 2<2n<m. ... However, since we*connect*in the following regulating*sets*with 3-face-*colorings*(for which we need plane graphs), we shall speak of regulating*sets*of plane graphs. ...##
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Universal Sets for Straight-Line Embeddings of Bicolored Graphs
[chapter]

2012
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Thirty Essays on Geometric Graph Theory
*

A

doi:10.1007/978-1-4614-0110-0_8
fatcat:g7zjupna3bhjxbbwaxl5agxrmy
*set*S of n*points*is 2-*color*universal for a graph G on n vertices if for every proper 2-*coloring*of G and for every 2-*coloring*of S with the same sizes of*color*classes as G has, G is straight-line ... We show that the so-called double chain is 2-*color*universal for paths if each of the two chains contains at least one fifth of all the*points*, but not if one of the chains is more than approximately 28 ... The main result of this paper is that some*point**sets*contain an NHAP for any equitable 2-*coloring*of their*points*. We call such*point**sets*2-*color*universal for a path. ...##
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Descriptive chromatic numbers of locally finite and everywhere two-ended graphs

2022
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Groups, Geometry, and Dynamics
*

We also provide a new bound for Borel chromatic numbers of graphs whose

doi:10.4171/ggd/643
fatcat:dhrlx6gzcvfvxdmppgoqciogwq
*connected*components all have two ends. ... Let D be the infinite*connected*component of G .C n N / not containing S . Let N 0 Â N be the*set*of elements of N adjacent to a*point*in D. Then, N 0 still divides C into 2 parts. ... Let ˆÂ OEG <1 be the*set*of*sets*which divide their*connected*component into two parts. ...##
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Descriptive Chromatic Numbers of Locally Finite and Everywhere Two Ended Graphs
[article]

2020
*
arXiv
*
pre-print

We also provide a new bound for Borel chromatic numbers of graphs whose

arXiv:2004.02316v1
fatcat:5nrakc3yave45jlmhlpjwccg5i
*connected*components all have two ends. ... Let D be the infinite*connected*component of G (C − N ) not containing S. Let N ⊂ N be the*set*of elements of N adjacent to a*point*in D. Then N still divides C into 2 parts. ... This follows from the fact that any T ∈ Ψ with T ⊂ C must be T -*connected*to S. Let N be the*set*of*points*in C + whose path distance from S is exactly 4. N is finite since G is locally finite. ...##
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Efficient Colorization of Large-Scale Point Cloud Using Multi-pass Z-Ordering

2014
*
2014 2nd International Conference on 3D Vision
*

We introduce a multi-pass Zordering technique that efficiently defines a graph structure to a large-scale and un-ordered

doi:10.1109/3dv.2014.33
dblp:conf/3dim/ChoYMB14
fatcat:cbqb7uc6une7pflhycfa5na4hi
*set*of 3D*points*, and use the graph structure for optimizing the*point**colors*to ... We present an efficient*colorization*method for a large scale*point*cloud using multi-view images. ... Although k-NN graph with k = 3 yields a*connected*graph, k may need to be a large value to guarantee a*connected*graph due to a large amount of*points*in more realistic*settings*. ...##
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Complexity and Computation of Connected Zero Forcing
[article]

2016
*
arXiv
*
pre-print

We show that the problem remains NP-hard when the initially

arXiv:1607.00658v1
fatcat:eqjlvk4n2retpays2dze7ic3oi
*colored**set*induces a*connected*subgraph. ... It is NP-hard to find a minimum zero forcing*set*- a smallest*set*of initially*colored*vertices which forces the entire graph to be*colored*. ... At this*point*, the*set*of*colored*vertices contains R, and can therefore*color*all of G; this means Z(G) ≤ |Z| < |R| = Z c (G) -a contradiction. ...##
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Page 29 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 41, Issue 1
[page]

1935
*
American Mathematical Society. Bulletin of the American Mathematical Society
*

The locus of such

*points*with respect to A is proved to be a linear graph m which, if S is simply*connected*, is a tree when S is closed and a*set*of infinite trees when S is open. ... Whyburn: Concerning the*connectivity*of limiting*sets*. In a compact metric space let the sequence of closed*sets*[A,,] converge to the limiting*set*A. ...
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