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

Fabrizio Frati, Marc Glisse, William J. Lenhart, Giuseppe Liotta, Tamara Mchedlidze, Rahnuma Islam Nishat
2013 Lecture Notes in Computer Science  
In this paper we study bichromatic 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.  ... 
doi:10.1007/978-3-642-36763-2_26 fatcat:vapgn5bqv5f2pp3rovqpsdasaa

Planar embedding of trees on point sets without the general position assumption

Asghar ASGHARIAN SARDROUD, Alireza BAGHERI
2015 Turkish Journal of Mathematics  
In this paper, we show that deciding the 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] .  ... 
doi:10.3906/mat-1410-12 fatcat:kc4egztq4bb5hmotl2g6czatvy

Open problems on graphs arising from geometric topology

Alberto Cavicchioli, Dušan Repovš, Arkadij B. Skopenkov
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 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".  ... 
doi:10.1016/s0166-8641(97)00093-x fatcat:bzzjj2sczjfg3du2kvcrrj3bgu

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)|).  ... 

On the Hardness of Point-Set Embeddability [chapter]

Stephane Durocher, Debajyoti Mondal
2012 Lecture Notes in Computer Science  
The problem 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] .  ... 
doi:10.1007/978-3-642-28076-4_16 fatcat:tesnzegtijdgzjgierj6bfsfni

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.  ... 

Embedding into Rectilinear Spaces

H. -J. Bandelt, V. Chepoi, M. Laurent
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 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.  ... 
doi:10.1007/pl00009370 fatcat:pfcoflckv5gozp3m5djlxudbvq

The game coloring number of pseudo partial k-trees

Xuding Zhu
2000 Discrete Mathematics  
By using this result, we prove that the game 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.  ... 
doi:10.1016/s0012-365x(99)00237-x fatcat:a55yzwogxfgefmhcbsttpre6ie

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.  ... 

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.  ... 

ON METRIC RAMSEY-TYPE DICHOTOMIES

YAIR BARTAL, NATHAN LINIAL, MANOR MENDEL, ASSAF NAOR
2005 Journal of the London Mathematical Society  
In particular, we show that a phase transition occurs at D=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.  ... 
doi:10.1112/s0024610704006155 fatcat:indjgmfxobgqloec7rntyyygb4

Graph minor theory

László Lovász
2005 Bulletin of the American Mathematical Society  
in a 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.  ... 
doi:10.1090/s0273-0979-05-01088-8 fatcat:nwz23353z5bubdsyobrdls3ksu

Simultaneous Embeddability of Two Partitions [article]

Jan Christoph Athenstädt, Tanja Hartmann, Martin Nöllenburg
2014 arXiv   pre-print
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 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.  ... 
arXiv:1408.6019v1 fatcat:ffxzjrahcbdndalpqko45c3yzi

ON EMBEDDING A GRAPH ON TWO SETS OF POINTS

EMILIO DI GIACOMO, GIUSEPPE LIOTTA, FRANCESCO TROTTA
2006 International Journal of Foundations of Computer Science  
Duranti 93, ABSTRACT Let R and B be two 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.  ... 
doi:10.1142/s0129054106004273 fatcat:b5d4usgomrfzfadm5is74aumxi
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