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### Connecting colored point sets

Oswin Aichholzer, Franz Aurenhammer, Thomas Hackl, Clemens Huemer
2007 Discrete Applied Mathematics
Let S = B ∪ R be a two-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] .  ...

Carmen Cortés, Alberto Márquez, Atsuhiro Nakamoto, Jesus Valenzuela
2005 European Workshop on Computational Geometry
Any metric quadrangulation (made by segments of straight line) of a 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.  ...

### Colored Point-set Embeddings of Acyclic Graphs [article]

Emilio Di Giacomo, Leszek Gasieniec, Giuseppe Liotta, Alfredo Navarra
2017 arXiv   pre-print
The lower bound holds even when the function that maps vertices to 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] .  ...

### Colored Point-Set Embeddings of Acyclic Graphs [chapter]

Emilio Di Giacomo, Leszek Gasieniec, Giuseppe Liotta, Alfredo Navarra
2018 Lecture Notes in Computer Science
The lower bound holds even when the function that maps vertices to 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] .  ...

### Combinatorial Species and Cluster Expansions

W. Faris
2010 Moscow Mathematical Journal
This paper will survey recent progress on clarifying the 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.  ...

### On the enumeration of finite maximal connected topologies

Shawpawn Kumar Das
1973 Journal of combinatorial theory. Series B (Print)
In this paper, it is shown that the number of homeomorphism classes of maximal 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).  ...

### Combinatorics and cluster expansions

William G. Faris
2010 Probability Surveys
This article is about the 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  ...

### On Embeddability of Buses in Point Sets [article]

Till Bruckdorfer and Michael Kaufmann and Stephen Kobourov and Sergey Pupyrev
2015 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  ...

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

Herbert Fleischner
Let G be a 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.  ...

### Universal Sets for Straight-Line Embeddings of Bicolored Graphs [chapter]

Josef Cibulka, Jan Kynčl, Viola Mészáros, Rudolf Stolař, Pavel Valtr
2012 Thirty Essays on Geometric Graph Theory
A 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.  ...

### Descriptive chromatic numbers of locally finite and everywhere two-ended graphs

Felix Weilacher
2022 Groups, Geometry, and Dynamics
We also provide a new bound for Borel chromatic numbers of graphs whose 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.  ...

### Descriptive Chromatic Numbers of Locally Finite and Everywhere Two Ended Graphs [article]

Felix Weilacher
2020 arXiv   pre-print
We also provide a new bound for Borel chromatic numbers of graphs whose 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.  ...

### Efficient Colorization of Large-Scale Point Cloud Using Multi-pass Z-Ordering

Sunyoung Cho, Jizhou Yan, Yasuyuki Matsushita, Hyeran Byun
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 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.  ...

### Complexity and Computation of Connected Zero Forcing [article]

Boris Brimkov
2016 arXiv   pre-print
We show that the problem remains NP-hard when the initially 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.  ...

### 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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