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k-colored Point-set Embeddability of Outerplanar Graphs

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Henk Meijer, Francesco Trotta, Stephen K. Wismath
2008 Journal of Graph Algorithms and Applications  
k disjoint sets S0, · · · , S k−1 of points in the plane with |Vi| = |Si| (i ∈ {0, · · · , k − 1}).  ...  Particular attention is devoted to outerplanar graphs, for which lower and upper bounds on the number of bends in the drawings are established.  ...  Graph G is k-colored point-set embeddable if it admits a point-set embedding on every k-colored set compatible with G.  ... 
doi:10.7155/jgaa.00158 fatcat:2wq5c7kx45crtk45d4jzw572me

k-Colored Point-Set Embeddability of Outerplanar Graphs [chapter]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Henk Meijer, Francesco Trotta, Stephen K. Wismath
2007 Lecture Notes in Computer Science  
k disjoint sets S0, · · · , S k−1 of points in the plane with |Vi| = |Si| (i ∈ {0, · · · , k − 1}).  ...  Particular attention is devoted to outerplanar graphs, for which lower and upper bounds on the number of bends in the drawings are established.  ...  Graph G is k-colored point-set embeddable if it admits a point-set embedding on every k-colored set compatible with G. Our main results are as follows.  ... 
doi:10.1007/978-3-540-70904-6_31 fatcat:jf3jukzfrrhhhpxcor35j27rdi

Compact Grid Representation of Graphs [chapter]

José Cáceres, Carmen Cortés, Clara Isabel Grima, Masahiro Hachimori, Alberto Márquez, Raiji Mukae, Atsuhiro Nakamoto, Seiya Negami, Rafael Robles, Jesús Valenzuela
2012 Lecture Notes in Computer Science  
A graph G is said to be grid locatable if it admits a representation such that vertices are mapped to grid points and edges to line segments that avoid grid points but the extremes.  ...  Additionally G is said to be properly embeddable in the grid if it is grid locatable and the segments representing edges do not cross each other.  ...  Thus, a very important question in this kind of representation is to minimize the area needed to represent a given graph. A line of the grid Z Z × Z Z is a set {(x, k)/x ∈ Z Z} for k ∈ Z Z fixed.  ... 
doi:10.1007/978-3-642-34191-5_16 fatcat:tpdtata7w5eerh4nyics6zzb7e

Constrained Point-Set Embeddability of Planar Graphs [chapter]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Henk Meijer, Stephen Wismath
2009 Lecture Notes in Computer Science  
It is proved that: (i) If G is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E \ E have at most 4 bends each  ...  This paper starts the investigation of a constrained version of the point-set embeddability problem.  ...  A limited list of papers about the k-colored point set embeddability includes [1, 4, 5, 8, 11, 12, 13] .  ... 
doi:10.1007/978-3-642-00219-9_35 fatcat:rpmcokecjrauvf7sdsrndkwv54

Asymmetric directed graph coloring games

Stephan Dominique Andres
2009 Discrete Mathematics  
From these results we deduce upper bounds for the (a, b)coloring number of oriented outerplanar graphs and of orientations of graphs embeddable in a surface with bounded girth.  ...  We prove that the (a, b)chromatic and (a, b)-coloring number for the class of orientations of forests is b + 2 if b ≤ a, and infinity otherwise.  ...  Fig. 1 . 1 The graph of Lemma 2 for (a, b) = (1, 2). Corollary 5 . 5 Let O be an orientation of an outerplanar graph, and a ≥ b.  ... 
doi:10.1016/j.disc.2008.03.022 fatcat:27yigtsq2zhg5fq7p3v7noavbq

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.  ...  Although the point-set embeddability problem is polynomial-time solvable for outerplanar graphs [4] and plane 3trees [16] , Cabello proved that it is NP-complete in general to decide whether a given  ... 
doi:10.1007/978-3-642-28076-4_16 fatcat:tesnzegtijdgzjgierj6bfsfni

Fast recognition of classes of almost-median graphs

Wilfried Imrich, Alenka Lipovec, Iztok Peterin, Petra Žigert
2007 Discrete Mathematics  
As a key auxiliary result we prove that all bipartite outerplanar graphs are isometric subgraphs of the hypercube and that the embedding can be effected in linear time.  ...  In this paper it is shown that a class of almost-median graphs that includes all planar almost-median graphs can be recognized in O(m log n) time, where n denotes the number of vertices and m the number  ...  A graph G is outerplanar if it is planar and embeddable into the plane such that all vertices lie on the outer face of the embedding. Such an embedding is called an outerplanar embedding of G.  ... 
doi:10.1016/j.disc.2006.07.011 fatcat:q777faqnf5hjfcrnvziltgflhq

Complete colorings of planar graphs

G. Araujo-Pardo, F.E. Contreras-Mendoza, S.J. Murillo-García, A.B. Ramos-Tort, C. Rubio-Montiel
2018 Discrete Applied Mathematics  
In this paper, we study the achromatic and the pseudoachromatic numbers of planar and outerplanar graphs as well as planar graphs of girth 4 and graphs embedded on a surface.  ...  We give asymptotically tight results and lower bounds for maximal embedded graphs.  ...  Introduction A k-coloring of a finite and simple graph G is a surjective function that assigns to each vertex of G a color from a set of k colors.  ... 
doi:10.1016/j.dam.2018.07.031 fatcat:gswsxjkouzggrpfrhifh7uvgcu

Bipartite minors

Maria Chudnovsky, Gil Kalai, Eran Nevo, Isabella Novik, Paul Seymour
2016 Journal of combinatorial theory. Series B (Print)  
We introduce a notion of bipartite minors and prove a bipartite analog of Wagner's theorem: a bipartite graph is planar if and only if it does not contain K 3,3 as a bipartite minor.  ...  Similarly, we provide a forbidden minor characterization for outerplanar graphs and forests.  ...  For example, G (5) is the barycentric subdivision of K 5 , endowed with a 2-coloring. Note that no edge of K 5 that connects two vertices of opposite colors is subdivided.  ... 
doi:10.1016/j.jctb.2015.08.001 fatcat:buoguul2tfc7np4lyy5fyxzzce

Boxicity of Graphs on Surfaces

Louis Esperet, Gwenaël Joret
2012 Graphs and Combinatorics  
In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface Σ of genus g is at most 5g+3.  ...  The boxicity of a graph G=(V,E) is the least integer k for which there exist k interval graphs G_i=(V,E_i), 1 < i < k, such that E=E_1 ∩ ... ∩ E_k.  ...  The intersection G 1 ∩ · · · ∩ G k of k graphs G 1 , . . . , G k defined on the same vertex set V is the graph (V, E 1 ∩ . . . ∩ E k ), where E i (1 i k) denotes the edge set of G i .  ... 
doi:10.1007/s00373-012-1130-x fatcat:arahokwfj5eevmjhzutkrgfq3y

Page 2013 of Mathematical Reviews Vol. 56, Issue 6 [page]

1978 Mathematical Reviews  
A finite linear space is a finite set of p elements called points together with a set of q sets of points called lines, such that each two points are on one and only one line and each line is on at least  ...  Finally, it is shown that colorability is a sort of complementary disconnectedness. S. F. Kapoor (Kalamazoo, Mich.) Mitchem, John 15475 A characterization of minimally non-outerplanar graphs. J.  ... 

Surfaces, Tree-Width, Clique-Minors, and Partitions

Guoli Ding, Bogdan Oporowski, Daniel P. Sanders, Dirk Vertigan
2000 Journal of combinatorial theory. Series B (Print)  
In 1971, Chartrand, Geller, and Hedetniemi conjectured that the edge set of a planar graph may be partitioned into two subsets, each of which induces an outerplanar graph.  ...  One such result, in which a planar graph may be thus edge partitioned into two series-parallel graphs, has nice generalizations for graphs embedded onto an arbitrary surface and graphs with no large clique-minor  ...  Let G be a graph embeddable on a surface 7 of Euler characteristic / . Let x be any vertex of G. Let V k be the set of vertices distance k from x. Let G k be the graph induced by V k .  ... 
doi:10.1006/jctb.2000.1962 fatcat:j7hw2spwajeaparmwn4uo4voae

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  ...  Finally, we show that universal convex point sets with n points exist for 1-bend bichromatic point-set embeddings of 2-colored trees. W. Didimo and M.  ...  [2] generalized the result by Pach and Wenger by proving that, for every k ≥ 2, a k-colored planar graph admits a k-colored point-set embedding on every k-coloured set of points with O(n) bends per  ... 
doi:10.1007/978-3-642-36763-2_26 fatcat:vapgn5bqv5f2pp3rovqpsdasaa

The game coloring number of pseudo partial k-trees

Xuding Zhu
2000 Discrete Mathematics  
, outerplanar graphs and planar graphs.  ...  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  ...  It was also proved in [8] that the game chromatic number of an outerplanar graph is at most 8.  ... 
doi:10.1016/s0012-365x(99)00237-x fatcat:a55yzwogxfgefmhcbsttpre6ie

Improved Bounds for Track Numbers of Planar Graphs [article]

Sergey Pupyrev
2019 arXiv   pre-print
A track layout of a graph consists of a vertex coloring and of a total order of each color class, such that the edges between each pair of colors form a non-crossing set.  ...  The track number of a graph is the minimum number of colors required by a track layout of the graph.  ...  We thank Jawaherul Alam, Michalis Bekos, Martin Gronemann, and Michael Kaufmann for fruitful initial discussions of the problem.  ... 
arXiv:1910.14153v1 fatcat:4dafhb3rgff5ra3ud6dx5i2kya
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