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Visibility Representations of Complete Graphs
[chapter]

1999
*
Lecture Notes in Computer Science
*

In this paper we study 3-dimensional

doi:10.1007/3-540-46648-7_34
fatcat:ndkw7jeloba2bac7lhyiuz3w4a
*visibility**representations**of**complete**graphs*. The vertices are represented by equal regular polygons lying in planes parallel to the xy-plane. ... We give the bounds for the maximal size f (k)*of*a clique represented by regular k-gons: k+1 2 + 2 ≤ f (k) ≤ 2 2 k and we present a particular result for triangles: f (3) ≥ 14. ... The basic idea is that the ordering*of*the polygons in these coordinates captures the geometric properties*of*our configuration*completely*. ...##
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3D Visibility Representations of Complete Graphs
[chapter]

2004
*
Lecture Notes in Computer Science
*

We give several bounds on the size

doi:10.1007/978-3-540-24595-7_21
fatcat:h5dcee73ojcszhcx5c6zessk54
*of*the largest*complete**graph*that has a 3D*visibility**representation*with particular properties. ... This paper continues the study*of*3D*visibility**representations**of**complete**graphs*where vertices are represented by equal convex polygons lying in planes parallel to the xy-plane. ... Introduction In this paper we study*visibility**representations**of**graphs*. ...##
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Rectangle-visibility representations of bipartite graphs
[chapter]

1995
*
Lecture Notes in Computer Science
*

The paper considers

doi:10.1007/3-540-58950-3_367
fatcat:wnpzivlpebemxd2fqhj2lqokra
*representations**of*bipartite*graphs*as rectanglevisibility*graphs*, i.e.,*graphs*whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical*visibility*... More generally, it is shown that Kp,q is a rectangle-*visibility**graph*if and only if p < 4. ... The*complete*and nearly*complete*bipartite*graphs*provide a source*of*examples for distinguishing among various subclasses*of*the class*of*thickness-two*graphs*. ...##
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On representations of some thickness-two graphs
[chapter]

1996
*
Lecture Notes in Computer Science
*

This paper considers

doi:10.1007/bfb0021815
fatcat:l35uyoigjngapl5llbrgsnlcpy
*representations**of**graphs*as rectanglevisibility*graphs*and as doubly linear*graphs*. ... These are, respectively,*graphs*whose vertices are isothetic rectangles in the plane with adjacency determined by horizontal and vertical*visibility*, and*graphs*that can be drawn as the union*of*two straight-edged ... Figures 1 and 2 show a rectangle-*visibility*and a doubly linear*representation*, respectively,*of*the*complete**graph*Ks. ...##
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A Visibility Representation for Graphs in Three Dimensions

1998
*
Journal of Graph Algorithms and Applications
*

We also note that planar

doi:10.7155/jgaa.00006
fatcat:dsb6xo6bvvfy5d5ikizwzaeyk4
*graphs*and*complete*bipartite*graphs*Km,n are*representable*, but that the family*of**representable**graphs*is not closed under*graph*minors. ... We apply an extension*of*the Erdős-Szekeres Theorem in a geometric setting to obtain an upper bound*of*n = 56 for the largest*representable**complete**graph*Kn. ...*Representations*for bipartite*graphs*As Figure 5 illustrates, every*complete*bipartite*graph*K m,n admits a z-*visibility**representation*. ...##
###
A Visibility Representation for Graphs in Three Dimensions
[chapter]

2002
*
Graph Algorithms and Applications I
*

We also note that planar

doi:10.1142/9789812777638_0006
fatcat:wbkh3oczorbvlazyduqk5nhujq
*graphs*and*complete*bipartite*graphs*Km,n are*representable*, but that the family*of**representable**graphs*is not closed under*graph*minors. ... We apply an extension*of*the Erdős-Szekeres Theorem in a geometric setting to obtain an upper bound*of*n = 56 for the largest*representable**complete**graph*Kn. ...*Representations*for bipartite*graphs*As Figure 5 illustrates, every*complete*bipartite*graph*K m,n admits a z-*visibility**representation*. ...##
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Further Results on Bar k-Visibility Graphs

2007
*
SIAM Journal on Discrete Mathematics
*

In this paper, we present sharp upper bounds on the maximum number

doi:10.1137/050644240
fatcat:4szbozjuofep5oc3bigpbrvaeq
*of*edges in a bar k-*visibility**graph*on n vertices and the largest order*of*a*complete*bar k-*visibility**graph*. ... A bar*visibility**representation**of*a*graph*G is a collection*of*horizontal bars in the plane corresponding to the vertices*of*G such that two vertices are adjacent if and only if the corresponding bars ... West for a helpful suggestion that simplified the statement*of*Proposition 7. ...##
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A unified approach to visibility representations of planar graphs

1986
*
Discrete & Computational Geometry
*

Every

doi:10.1007/bf02187705
fatcat:ztf5krfe3jazdm3ctck5ca3px4
*graph*that admits this*representation*must be planar. We consider three types*of**visibility**representations*, and we give*complete*characterizations*of*the classes*of**graphs*that admit them. ... Furthermore, we present linear time algorithms for testing the existence*of*and constructing*visibility**representations**of*planar*graphs*. Many applications*of*our results can be found in VLSI layout. ... Acknowledgments We wish to thank Franco Preparata for suggesting the connection between Hamiltonian paths and*visibility*, and Doug West for useful discussions. ...##
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Some new results on bar visibility of digraphs
[article]

2021
*
arXiv
*
pre-print

Discrete Math. 27(3) (2013) 1429–1449) as a natural generalization

arXiv:2108.12086v1
fatcat:f3zv77rwjvdxbdwzugeqcodssy
*of*t-bar*visibility**representation**of*undirected*graphs*. ...*Visibility**representation**of*digraphs was introduced by Axenovich, Beveridge, Hutchinson, and West (SIAM J. ... To*complete*the proof, we obtain a (k + 1)-bar*visibility**representation**of*T n+2 from a k-bar*visibility**representation**of*T n . ...##
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Bar k-Visibility Graphs: Bounds on the Number of Edges, Chromatic Number, and Thickness
[chapter]

2006
*
Lecture Notes in Computer Science
*

As a consequence, we obtain an upper bound

doi:10.1007/11618058_7
fatcat:bln3zi4ltjdrjdmjlutigtsmi4
*of*12 on the chromatic number*of*bar 1-*visibility**graphs*, and a tight upper bound*of*8 on the size*of*the largest*complete*bar 1-*visibility**graph*. ... We say that G is a bar*visibility**graph*, and S its bar*visibility**representation*, if there exists a one-to-one correspondence between vertices*of*G and bars in S, such that there is an edge between two ... K 4k+4 is the largest*complete*bar k-*visibility**graph*. Proof (Assuming Conjecture 1). Figure 6 shows a bar k-*visibility**representation**of*K 4k+4 . ...##
###
On representations of some thickness-two graphs

1999
*
Computational geometry
*

This paper considers

doi:10.1016/s0925-7721(99)00018-8
fatcat:5bcoo7cxgrcdxg2vnwffkx5rwi
*representations**of**graphs*as rectangle-*visibility**graphs*and as doubly linear*graphs*. ... These are, respectively,*graphs*whose vertices are isothetic rectangles in the plane with adjacency determined by horizontal and vertical*visibility*, and*graphs*that can be drawn as the union*of*two straight-edged ... Acknowledgements The authors would like to thank the participants in the Workshop on*Visibility**Representations*, held in February 1993, at the McGill University Bellairs Research Institute, Barbados, and ...##
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On a visibility representation of graphs
[chapter]

1996
*
Lecture Notes in Computer Science
*

We give a

doi:10.1007/bfb0021799
fatcat:6tsj7svmjrgxzmslvtcjbsgjt4
*visibility**representation**of**graphs*which extends some very well-known*representations*considered extensively in the literature. ... We consider this problem for concrete well-known families*of**graphs*such as planar*graphs*,*complete**graphs*and*complete*bipartite*graphs*. ...*Of*course, both, the objects and the*visibility*used play an important role in characterizing the types*of**graphs*that admit*visibility**representations*. ...##
###
On the number of directions in visibility representations of graphs (extended abstract)
[chapter]

1995
*
Lecture Notes in Computer Science
*

In such a

doi:10.1007/3-540-58950-3_368
fatcat:qo3mnpexyjhiha6v7p3eosmsga
*representation*, the given*graph*is decomposed into a union*of*unidirectional*visibility**graphs*, for the chosen set*of*directions. ... We consider*visibility**representations**of**graphs*in which the vertices are represented by a collection O*of*non-overlapping convex regions on the plane. ... Acknowledgements Many thanks to Mark Noy for suggesting the use*of*Lemma 8, Michel Pocchiola for useful conversations, and Roberto Tamassia for pointing out relevant references. ...##
###
Area Minimization for Grid Visibility Representation of Hierarchically Planar Graphs
[chapter]

1999
*
Lecture Notes in Computer Science
*

In this paper, we shall investigate the computational complexity

doi:10.1007/3-540-48686-0_9
fatcat:u3mbk5b4ejegpabyxbttnnhgmi
*of*constructing minimum area grid*visibility**representations**of*hierarchically planar*graphs*. ... Hierarchical*graphs*are an important class*of**graphs*for modelling many real applications in software and information visualization. ... In this paper we are focused only on a \*visibility**representation*"*of*a hierarchically planar*graph*. ...##
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Universal 3-dimensional visibility representations for graphs

1998
*
Computational geometry
*

This paper studies 3-dimensional

doi:10.1016/s0925-7721(97)00015-1
fatcat:fed4nmehsffynnlr246cyyg53u
*visibility**representations**of**graphs*in which objects in 3-d correspond to vertices and vertical*visibilities*between these objects correspond to edges. ... We also study the universality*of*classes*of*simple objects (translates*of*a single, not necessarily polygonal object) relative to cliques Kn and similarly relative to*complete*bipartite*graphs*K,~,m. ... From the point*of*view*of*geometric*graph*theory, it is natural to consider*visibility**representations**of**graphs*in dimensions higher than 2. ...
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