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Point-Set Embeddability of 2-Colored Trees
[chapter]

2013
*
Lecture Notes in Computer Science
*

In this paper we study bichromatic

doi:10.1007/978-3-642-36763-2_26
fatcat:vapgn5bqv5f2pp3rovqpsdasaa
*point*-*set*embeddings*of**2*-*colored**trees*on*2*-*colored**point**sets*, i.e.,*point*-*set*embeddings*of**trees*(whose vertices are*colored*red and blue) on*point**sets*(whose*points*...*of**2*-*colored**trees*(resp.*2*-*colored*binary*trees*). ... the*point*-*set**embeddability**of**2*-*colored**trees*on*2*-*colored**point**sets*. ...##
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Planar embedding of trees on point sets without the general position assumption

2015
*
Turkish Journal of Mathematics
*

In this paper, we show that deciding the

doi:10.3906/mat-1410-12
fatcat:kc4egztq4bb5hmotl2g6czatvy
*point*-*set**embeddability**of**trees*without the general position assumption is NP-complete. ... Then we introduce an algorithm for*point*-*set*embedding*of*n -node binary*trees*with at most n 3 total bends on any*point**set*. ... In*colored*embedding, the nodes*of*the given graph and the given*points*are*colored*and each node should be mapped on a*point**of*the same*color*[1, 7] . ...##
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Open problems on graphs arising from geometric topology

1998
*
Topology and its Applications
*

We have collected several open problems on graphs which arise in geometric topology, in particular in the following areas: (1) basic

doi:10.1016/s0166-8641(97)00093-x
fatcat:bzzjj2sczjfg3du2kvcrrj3bgu
*embeddability**of*compacta into the plane R'; (*2*) approximability*of*... * and their span; and (4) representations*of*closed PL manifolds by*colored*graphs. These problems should be*of*interest to both topologists and combinatorists. 0 1998 Elsevier Science B.V. ... Take a*point**2*E 5". ...##
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Page 4145 of Mathematical Reviews Vol. , Issue 82j
[page]

1982
*
Mathematical Reviews
*

Pick a

*set**of*r*points*in S, and partition this*set*into ¢ nonempty classes. If the*points*in each class are identified, the resulting object is called a pseudosurface or pinched manifold [see P. C. ... Let 7(G) denote the*set**of*all spanning*trees*for the connected graph G, and let C(H) denote the*set**of*components*of*an arbitrary graph H (with c(H)=|C(H)|). ...##
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On the Hardness of Point-Set Embeddability
[chapter]

2012
*
Lecture Notes in Computer Science
*

The problem

doi:10.1007/978-3-642-28076-4_16
fatcat:tesnzegtijdgzjgierj6bfsfni
*of*deciding whether a plane graph admits a*point*-*set*embedding on a given*set**of**points*is NPcomplete for*2*-connected planar graphs, but polynomial-time solvable for outerplanar graphs and ... A*point*-*set*embedding*of*a plane graph G with n vertices on a*set*S*of*n*points*is a straight-line drawing*of*G, where the vertices*of*G are mapped to distinct*points**of*S. ...*2*-connected plane graph admits a*point*-*set*embedding on a given*set**of**points*[5] . ...##
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Page 22 of Mathematical Reviews Vol. 55, Issue 1
[page]

1978
*
Mathematical Reviews
*

Number Theory 8 (1976), no.

*2*, 137-156. A connected planar graph is called modular if it is vertex two*colorable*so that vertices*of**color*A have degree | or*2*and those*of**color*B degree 1 or 3. ... Using this notion,*sets**of*classes*of*isomorphic*trees*are defined whose invariant vectors bear a certain relation to given vectors q and r. The complement and dual*trees**of*a*tree*are defined. ...##
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Embedding into Rectilinear Spaces

1998
*
Discrete & Computational Geometry
*

We show that the problem whether a given finite metric space (X, d) can be embedded into the rectilinear space R m can be formulated in terms

doi:10.1007/pl00009370
fatcat:pfcoflckv5gozp3m5djlxudbvq
*of*m-*colorability**of*a certain hypergraph associated with ( ... We also consider the question*of*determining the maximum number*of*equidistant*points*that can be placed in the m-dimensional rectilinear space and show that this number is equal to 2m for m ≤ 3. 596 H ... Acknowledgment We thank the referees for providing the references to the work*of*Bonan-Hamada and Guy. ...##
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The game coloring number of pseudo partial k-trees

2000
*
Discrete Mathematics
*

By using this result, we prove that the game

doi:10.1016/s0012-365x(99)00237-x
fatcat:a55yzwogxfgefmhcbsttpre6ie
*coloring*number*of*a graph*embeddable*on an orientable surface*of*genus g¿1 is at most 1*2*(3 √ 1 + 48g + 23) : This is the ÿrst upper bound for the game*coloring*... Namely, we prove that the game*coloring*number*of*an (a; b)-pseudo partial*2*-*tree*is at most a + b + 8. ... Let G(S) be the*set**of*graphs*of*minimum degree ¿*2*and*embeddable*on S. Then it is easy to see that G(S) satisÿes (1), (*2*), (4) and (5)*of*Deÿnition 3. ...##
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Page 6593 of Mathematical Reviews Vol. , Issue 97K
[page]

1997
*
Mathematical Reviews
*

Summary: “A

*tree*decomposition*of*a graph G is a family*of*subtrees whose*sets**of*edges partition the*set**of*edges*of*G. The minimum number*of**trees*in a*tree*decomposition is the*tree*number*of*G. ... In this pa- per we prove that if |A| < |3(m —*2*)/*2*| then D is*embeddable*and, moreover, there is a complementing permutation which is a com- position*of*disjoint transpositions when n is even or a composition ...##
###
Page 6813 of Mathematical Reviews Vol. , Issue 2000j
[page]

2000
*
Mathematical Reviews
*

A directed forest or oriented forest is a disjoint union

*of*directed graphs, each*of*which has a rooted*tree*as its underlying undi- rected subgraph, with each edge*pointing*away from the root. ... The complex A(G,,), where G, is the complete bidirected graph on n vertices, is the basis for Pitman’s elegant proof*of*Cayley’s formula n"~' for the number*of*rooted*trees*on an n-element*set*[see J. ...##
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Page 33 of Mathematical Reviews Vol. 51, Issue 1
[page]

1976
*
Mathematical Reviews
*

Let T be a

*tree*, let {v,: i=1,*2*, - - -, m} be the*set**of*all*points**of*T whose degrees are greater than 3, and let {u,: j=1,*2*, ---, n} be the*set**of*all*points**of*T whose degrees are |. ... For any*tree*T the minimum number*of*line disjoint paths in T is half*of*the number*of**points**of*odd degree in T [see R. G. Busacker and T. L. ...##
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ON METRIC RAMSEY-TYPE DICHOTOMIES

2005
*
Journal of the London Mathematical Society
*

In particular, we show that a phase transition occurs at D=

doi:10.1112/s0024610704006155
fatcat:indjgmfxobgqloec7rntyyygb4
*2*. ... The classical Ramsey theorem, states that every graph contains either a large clique or a large independent*set*. ... The extension*of*this upper bound for E k (α, n), k ≥ 1, α ∈ (1,*2*), is contained in Proposition 12. ...##
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Graph minor theory

2005
*
Bulletin of the American Mathematical Society
*

in a

doi:10.1090/s0273-0979-05-01088-8
fatcat:nwz23353z5bubdsyobrdls3ksu
*tree*-like fashion from graphs that can almost be embedded in a fixed surface. ... The motivating problem was Kuratowski's characterization*of*planar graphs, and a far-reaching generalization*of*this, conjectured by Wagner: If a class*of*graphs is minor-closed (i.e., it is closed under ... [*2*] ). Let the chromatic number serve as an example: this is defined as the minimum number*of**colors*needed to*color*the nodes*of*a graph G so that adjacent nodes get different*colors*. ...##
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Simultaneous Embeddability of Two Partitions
[article]

2014
*
arXiv
*
pre-print

Each element

arXiv:1408.6019v1
fatcat:ffxzjrahcbdndalpqko45c3yzi
*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 study the simultaneous*embeddability**of*a pair*of*partitions*of*the same underlying*set*into disjoint blocks. ... Any two partitions*of*a common universe are weakly*embeddable*on any*point**set*. Proof. ...##
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ON EMBEDDING A GRAPH ON TWO SETS OF POINTS

2006
*
International Journal of Foundations of Computer Science
*

Duranti 93, ABSTRACT Let R and B be two

doi:10.1142/s0129054106004273
fatcat:b5d4usgomrfzfadm5is74aumxi
*sets**of**points*such that the*points**of*R are*colored*red and the*points**of*B are*colored*blue. ... We also prove that two bends per edge suffice if G is a*2*-*colored*caterpillar and that for properly*2*-*colored*caterpillars, properly*2*-*colored*wreaths,*2*-*colored*paths, and*2*-*colored*cycles the number ... For example, are all*2*-*colored**trees*bi-chromatic*point*-*set**embeddable*with a constant number*of*bends per edge? • Let G be a*2*-*colored*planar graph and let S be a*set**of**points*equipollent with G. ...
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