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k-colored Point-set Embeddability of Outerplanar Graphs

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. ...

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

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. ...

##
###
Compact Grid Representation of Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

A

doi:10.1007/978-3-642-34191-5_16
fatcat:tpdtata7w5eerh4nyics6zzb7e
*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. ...##
###
Constrained Point-Set Embeddability of Planar Graphs
[chapter]

2009
*
Lecture Notes in Computer Science
*

It is proved that: (i) If G is an

doi:10.1007/978-3-642-00219-9_35
fatcat:rpmcokecjrauvf7sdsrndkwv54
*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] . ...##
###
Asymmetric directed graph coloring games

2009
*
Discrete Mathematics
*

From these results we deduce upper bounds for the (a, b)

doi:10.1016/j.disc.2008.03.022
fatcat:27yigtsq2zhg5fq7p3v7noavbq
*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. ...##
###
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. ... 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 ...##
###
Fast recognition of classes of almost-median graphs

2007
*
Discrete Mathematics
*

As a key auxiliary result we prove that all bipartite

doi:10.1016/j.disc.2006.07.011
fatcat:q777faqnf5hjfcrnvziltgflhq
*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. ...##
###
Complete colorings of planar graphs

2018
*
Discrete Applied Mathematics
*

In this paper, we study the achromatic and the pseudoachromatic numbers

doi:10.1016/j.dam.2018.07.031
fatcat:gswsxjkouzggrpfrhifh7uvgcu
*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*. ...##
###
Bipartite minors

2016
*
Journal of combinatorial theory. Series B (Print)
*

We introduce a notion

doi:10.1016/j.jctb.2015.08.001
fatcat:buoguul2tfc7np4lyy5fyxzzce
*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. ...##
###
Boxicity of Graphs on Surfaces

2012
*
Graphs and Combinatorics
*

In this note we prove that the boxicity

doi:10.1007/s00373-012-1130-x
fatcat:arahokwfj5eevmjhzutkrgfq3y
*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 . ...##
###
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

2000
*
Journal of combinatorial theory. Series B (Print)
*

In 1971, Chartrand, Geller, and Hedetniemi conjectured that the edge

doi:10.1006/jctb.2000.1962
fatcat:j7hw2spwajeaparmwn4uo4voae
*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*. ...##
###
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*... 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 ...##
###
The game coloring number of pseudo partial k-trees

2000
*
Discrete Mathematics
*

,

doi:10.1016/s0012-365x(99)00237-x
fatcat:a55yzwogxfgefmhcbsttpre6ie
*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. ...##
###
Improved Bounds for Track Numbers of Planar Graphs
[article]

2019
*
arXiv
*
pre-print

A track layout

arXiv:1910.14153v1
fatcat:4dafhb3rgff5ra3ud6dx5i2kya
*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. ...
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