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Drawing Colored Graphs with Constrained Vertex Positions and Few Bends per Edge [chapter]

Emilio Di Giacomo, Giuseppe Liotta, Francesco Trotta
2008 Lecture Notes in Computer Science  
Hamiltonicity, book embeddability, and point-set embeddability of planar graphs are strictly related concepts.  ...  have O(k) bends per edge.  ...  most h; and (iii) the bends of Γ are at grid points.  ... 
doi:10.1007/978-3-540-77537-9_31 fatcat:xw4gximprrb6tkgbsbilgenzyy

Drawing Colored Graphs with Constrained Vertex Positions and Few Bends per Edge

Emilio Di Giacomo, Giuseppe Liotta, Francesco Trotta
2008 Algorithmica  
Hamiltonicity, book embeddability, and point-set embeddability of planar graphs are strictly related concepts.  ...  have O(k) bends per edge.  ...  most h; and (iii) the bends of Γ are at grid points.  ... 
doi:10.1007/s00453-008-9255-2 fatcat:ahnv4ewbt5f3pnigx53bfke25y

A New Algorithm for Embedding Plane Graphs at Fixed Vertex Locations

Marcus Schaefer
2021 Electronic Journal of Combinatorics  
We show that a plane graph can be embedded with its vertices at arbitrarily assigned locations in the plane and at most $6n-1$ bends per edge.  ...  This improves and simplifies a classic result by Pach and Wenger. The proof extends to orthogonal drawings.  ...  Acknowledgement I would like to thank the anonymous referees for various suggestions that improved the paper, but particularly for spotting an error in the proof of Lemma 7, and for suggesting grid embeddings  ... 
doi:10.37236/10106 fatcat:hjfxxpg745fwbitreq573gq6ey

Smooth Orthogonal Layouts

Michael A. Bekos, Michael Kaufmann, Stephen G. Kobourov, Antonios Symvonis
2013 Journal of Graph Algorithms and Applications  
While in traditional orthogonal layouts every edge is made of a sequence of axis-aligned line segments, in smooth orthogonal layouts every edge is made of axis-aligned segments and circular arcs with common  ...  Our goal is to create such layouts with low edge complexity, measured by the number of line and circular arc segments.  ...  Nöllenburg and I. Rutter for helpful discussions.  ... 
doi:10.7155/jgaa.00305 fatcat:cpyhzz7ks5gqrim3xsxpu3nqpy

Smooth Orthogonal Layouts [chapter]

Michael A. Bekos, Michael Kaufmann, Stephen G. Kobourov, Antonios Symvonis
2013 Lecture Notes in Computer Science  
While in traditional orthogonal layouts every edge is made of a sequence of axis-aligned line segments, in smooth orthogonal layouts every edge is made of axis-aligned segments and circular arcs with common  ...  Our goal is to create such layouts with low edge complexity, measured by the number of line and circular arc segments.  ...  Nöllenburg and I. Rutter for helpful discussions.  ... 
doi:10.1007/978-3-642-36763-2_14 fatcat:5zg4c4omgrekrmuocaklaaqxha

A note on simultaneous embedding of planar graphs

Emilio Di Giacomo, Giuseppe Liotta
2005 European Workshop on Computational Geometry  
Erten and Kobourov show that the overall time complexity of the procedure is O(n) where n is the number of vertices in G 1 and in G 2 ; also, if the bends may not be at integer grid points, the size of  ...  an augmented Hamiltonian cycle in the tree with the additional constraint that the cycle must contain all edges of P .  ...  G and P can be simultaneously embedded with fixed edges in O(n) time, using at most one bend for each edge of G and zero bends for each edge of P , on an integer grid of size O(n) × O(n 2 ), where n =  ... 
dblp:conf/ewcg/GiacomoL05 fatcat:7cidzaxv5vhnniu7v43kq6nc3m

Radial Drawings of Graphs: Geometric Constraints and Trade-Offs [chapter]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta
2007 Lecture Notes in Computer Science  
The following requirements are considered: vertex centrality, edge crossings, curve complexity, and vertex radial distribution.  ...  This paper studies how to compute radial drawings of graphs by taking into account additional geometric constraints which correspond to typical aesthetic and semantic requirements for the visualization  ...  that is optimal in terms of crossings and curve complexity and radial distribution.  ... 
doi:10.1007/978-3-540-70904-6_34 fatcat:jteieyrqrjccda2bklup723qdi

The Hamiltonian Path Graph is Connected for Simple s,t Paths in Rectangular Grid Graphs [article]

Rahnuma Islam Nishat, Venkatesh Srinivasan, Sue Whitesides
2022 arXiv   pre-print
A simple s,t path P in a rectangular grid graph 𝔾 is a Hamiltonian path from the top-left corner s to the bottom-right corner t such that each internal subpath of P with both endpoints a and b on the  ...  Our reconfiguration result proves that the Hamiltonian path graph G for simple s,t paths is connected and has diameter at most 5|𝔾|/4 which is asymptotically tight.  ...  They used two operations flip and transpose, and showed that the Hamiltonian cycle graph with respect to those two operations is connected for 1-complex cycles in rectangular grids and L-shaped grid graphs  ... 
arXiv:2205.08025v1 fatcat:akseq5h7tjbb5hgymx2gjxeuam

Radial drawings of graphs: Geometric constraints and trade-offs

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta
2008 Journal of Discrete Algorithms  
The following requirements are considered: vertex centrality, edge crossings, curve complexity, and radial distribution of the vertices.  ...  This paper studies how to compute radially layered drawings of graphs by taking into account additional geometric constraints which correspond to typical aesthetic and semantic requirements for the visualization  ...  A graph G is Hamiltonian if it has a simple cycle that contains all its vertices; such a cycle is called a Hamiltonian cycle of G. Suppose that G is planar and that G is not Hamiltonian.  ... 
doi:10.1016/j.jda.2006.12.007 fatcat:rfk2r5l42nhavncw4cixt2c5au

Embedding Planar Graphs at Fixed Vertex Locations [chapter]

János Pach, Rephael Wenger
1998 Lecture Notes in Computer Science  
Let G be a planar graph of n vertices, v1, . . . , vn, and let {p1, . . . , pn} be a set of n points in the plane.  ...  In fact, if G is a planar graph containing at least m pairwise independent edges and the vertices of G are randomly assigned to points in convex position, then, almost surely, every planar embedding of  ...  Every planar graph containing a Hamiltonian cycle can be divided into two outerplanar graphs which have only the edges of the Hamiltonian cycle in common.  ... 
doi:10.1007/3-540-37623-2_20 fatcat:hbycpmuhfbcbzkl3clkafpyrs4

Manhattan-Geodesic Embedding of Planar Graphs [chapter]

Bastian Katz, Marcus Krug, Ignaz Rutter, Alexander Wolff
2010 Lecture Notes in Computer Science  
In contrast, we efficiently solve geodesic polygonization-the special case where the graph is a cycle.  ...  In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention.  ...  AW thanks Ferran Hurtado for inviting him to a workshop in 2006, Manuel Abellanas for bringing Labeled Geodesic PSE to his attention there, and Stefan Langerman and Pat Morin for discussions about the  ... 
doi:10.1007/978-3-642-11805-0_21 fatcat:2htc4dqdlralpe5m5bc34bmomy

Simultaneous Embedding of Planar Graphs [article]

Thomas Bläsius and Stephen G. Kobourov and Ignaz Rutter
2015 arXiv   pre-print
This forms the basis for the visualization of dynamic graphs and thus is an important field of research.  ...  Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices.  ...  [31, 32] to at most two bends per edge in general and one bend per edge, if G 1 and G 2 are both sub-Hamiltonian.  ... 
arXiv:1204.5853v3 fatcat:op5qpoub7bfhlgljxqygzx6jha

Not being (super)thin or solid is hard: A study of grid Hamiltonicity

Esther M. Arkin, Sándor P. Fekete, Kamrul Islam, Henk Meijer, Joseph S.B. Mitchell, Yurai Núñez-Rodríguez, Valentin Polishchuk, David Rappaport, Henry Xiao
2009 Computational geometry  
For many classes of grid graphs we resolve the computational complexity of the Hamiltonian cycle problem.  ...  As applications of the correspondence, we show that for graphs in C g the Hamiltonian cycle problem is NP-complete and that for any N 5 there exist graphs in C g that have exactly N Hamiltonian cycles.  ...  Polishchuk is supported in part by Academy of Finland grant 118653 (ALGODAN).  ... 
doi:10.1016/j.comgeo.2008.11.004 fatcat:is4zzl2opvgapi4tn6rovmsdlm

Vertex Intersection Graphs of Paths on a Grid

Andrei Asinowski, Elad Cohen, Martin Charles Golumbic, Vincent Limouzy, Marina Lipshteyn, Michal Stern
2012 Journal of Graph Algorithms and Applications  
We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns).  ...  The grid intersection graphs are shown to be equivalent to the bipartite B 0 -VPG graphs and the circle graphs are strictly contained in B 1 -VPG.  ...  This method has O(n + m) time complexity where n is the number of vertices in the graph (segments) and m is the number of crosspoints and internal points on the grid.  ... 
doi:10.7155/jgaa.00253 fatcat:txmeh3fdxfgghhx4dguptafhri

The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes

Maciej Liśkiewicz, Mitsunori Ogihara, Seinosuke Toda
2003 Theoretical Computer Science  
Comput. 8 (1979) 410 -421) showed that the problem of computing the number of simple s-t paths in graphs is #P-complete both in the case of directed graphs and in the case of undirected graphs.  ...  Welsh (Complexity: Knots, Colourings and Counting, Cambridge University Press, Cambridge, 1993, p. 17) asked whether the problem of computing the number of self-avoiding walks of a given length in the  ...  Acknowledgements The authors are grateful to J org Rothe for his helpful comments and his careful reading of a draft to detect errors in an early version of the paper and to Andreas Jakoby and Dick Lipton  ... 
doi:10.1016/s0304-3975(03)00080-x fatcat:vybthjqqabaapbpsdgeftyuedq
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