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Visibility Representations of Complete Graphs [chapter]

Robert Babilon, Helena Nyklová, Ondřej Pangrác, Jan Vondrák
1999 Lecture Notes in Computer Science  
In this paper we study 3-dimensional 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.  ... 
doi:10.1007/3-540-46648-7_34 fatcat:ndkw7jeloba2bac7lhyiuz3w4a

3D Visibility Representations of Complete Graphs [chapter]

Jan Štola
2004 Lecture Notes in Computer Science  
We give several bounds on the size 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.  ... 
doi:10.1007/978-3-540-24595-7_21 fatcat:h5dcee73ojcszhcx5c6zessk54

Rectangle-visibility representations of bipartite graphs [chapter]

Alice M. Dean, Joan P. Hutchinson
1995 Lecture Notes in Computer Science  
The paper considers 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.  ... 
doi:10.1007/3-540-58950-3_367 fatcat:wnpzivlpebemxd2fqhj2lqokra

On representations of some thickness-two graphs [chapter]

Joan P. Hutchinson, Thomas Shermer, Andrew Vince
1996 Lecture Notes in Computer Science  
This paper considers 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.  ... 
doi:10.1007/bfb0021815 fatcat:l35uyoigjngapl5llbrgsnlcpy

A Visibility Representation for Graphs in Three Dimensions

Prosenjit Bose, Hazel Everett, Sándor P. Fekete, Michael E. Houle, Anna Lubiw, Henk Meijer, Kathleen Romanik, Günter Rote, Thomas C. Shermer, Sue Whitesides, Christian Zelle
1998 Journal of Graph Algorithms and Applications  
We also note that planar 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.  ... 
doi:10.7155/jgaa.00006 fatcat:dsb6xo6bvvfy5d5ikizwzaeyk4

A Visibility Representation for Graphs in Three Dimensions [chapter]

Prosenjit Bose, Hazel Everett, Sándor P. Fekete, Michael E. Houle, Anna Lubiw, Henk Meijer, Kathleen Romanik, Günter Rote, Thomas C. Shermer, Sue Whitesides, Christian Zelle
2002 Graph Algorithms and Applications I  
We also note that planar 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.  ... 
doi:10.1142/9789812777638_0006 fatcat:wbkh3oczorbvlazyduqk5nhujq

Further Results on Bar k-Visibility Graphs

Stephen G. Hartke, Jennifer Vandenbussche, Paul Wenger
2007 SIAM Journal on Discrete Mathematics  
In this paper, we present sharp upper bounds on the maximum number 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.  ... 
doi:10.1137/050644240 fatcat:4szbozjuofep5oc3bigpbrvaeq

A unified approach to visibility representations of planar graphs

Roberto Tamassia, Ioannis G. Tollis
1986 Discrete & Computational Geometry  
Every 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.  ... 
doi:10.1007/bf02187705 fatcat:ztf5krfe3jazdm3ctck5ca3px4

Some new results on bar visibility of digraphs [article]

Yuanrui Feng, Jun Ge, Douglas B. West, Yan Yang
2021 arXiv   pre-print
Discrete Math. 27(3) (2013) 1429–1449) as a natural generalization 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 .  ... 
arXiv:2108.12086v1 fatcat:f3zv77rwjvdxbdwzugeqcodssy

Bar k-Visibility Graphs: Bounds on the Number of Edges, Chromatic Number, and Thickness [chapter]

Alice M. Dean, William Evans, Ellen Gethner, Joshua D. Laison, Mohammad Ali Safari, William T. Trotter
2006 Lecture Notes in Computer Science  
As a consequence, we obtain an upper bound 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 .  ... 
doi:10.1007/11618058_7 fatcat:bln3zi4ltjdrjdmjlutigtsmi4

On representations of some thickness-two graphs

Joan P. Hutchinson, Thomas Shermer, Andrew Vince
1999 Computational geometry  
This paper considers 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  ... 
doi:10.1016/s0925-7721(99)00018-8 fatcat:5bcoo7cxgrcdxg2vnwffkx5rwi

On a visibility representation of graphs [chapter]

F. J. Cobos, J. C. Dana, F. Hurtado, A. Márquez, F. Mateos
1996 Lecture Notes in Computer Science  
We give a 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.  ... 
doi:10.1007/bfb0021799 fatcat:6tsj7svmjrgxzmslvtcjbsgjt4

On the number of directions in visibility representations of graphs (extended abstract) [chapter]

Evangelos Kranakis, Danny Krizanc, Jorge Urrutia
1995 Lecture Notes in Computer Science  
In such a 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.  ... 
doi:10.1007/3-540-58950-3_368 fatcat:qo3mnpexyjhiha6v7p3eosmsga

Area Minimization for Grid Visibility Representation of Hierarchically Planar Graphs [chapter]

Xuemin Lin, Peter Eades
1999 Lecture Notes in Computer Science  
In this paper, we shall investigate the computational complexity 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.  ... 
doi:10.1007/3-540-48686-0_9 fatcat:u3mbk5b4ejegpabyxbttnnhgmi

Universal 3-dimensional visibility representations for graphs

Helmut Alt, Michael Godau, Sue Whitesides
1998 Computational geometry  
This paper studies 3-dimensional 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.  ... 
doi:10.1016/s0925-7721(97)00015-1 fatcat:fed4nmehsffynnlr246cyyg53u
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