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

2008
*
Lecture Notes in Computer Science
*

Hamiltonicity, book embeddability,

doi:10.1007/978-3-540-77537-9_31
fatcat:xw4gximprrb6tkgbsbilgenzyy
*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. ...##
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Drawing Colored Graphs with Constrained Vertex Positions and Few Bends per Edge

2008
*
Algorithmica
*

Hamiltonicity, book embeddability,

doi:10.1007/s00453-008-9255-2
fatcat:ahnv4ewbt5f3pnigx53bfke25y
*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. ...##
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A New Algorithm for Embedding Plane Graphs at Fixed Vertex Locations

2021
*
Electronic Journal of Combinatorics
*

We show that a plane

doi:10.37236/10106
fatcat:hjfxxpg745fwbitreq573gq6ey
*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 ...##
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Smooth Orthogonal Layouts

2013
*
Journal of Graph Algorithms and Applications
*

While

doi:10.7155/jgaa.00305
fatcat:cpyhzz7ks5gqrim3xsxpu3nqpy
*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. ...##
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Smooth Orthogonal Layouts
[chapter]

2013
*
Lecture Notes in Computer Science
*

While

doi:10.1007/978-3-642-36763-2_14
fatcat:5zg4c4omgrekrmuocaklaaqxha
*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. ...##
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A note on simultaneous embedding of planar graphs

2005
*
European Workshop on Computational Geometry
*

Erten

dblp:conf/ewcg/GiacomoL05
fatcat:7cidzaxv5vhnniu7v43kq6nc3m
*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 = ...##
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Radial Drawings of Graphs: Geometric Constraints and Trade-Offs
[chapter]

2007
*
Lecture Notes in Computer Science
*

The following requirements are considered: vertex centrality, edge crossings, curve

doi:10.1007/978-3-540-70904-6_34
fatcat:jteieyrqrjccda2bklup723qdi
*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. ...##
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The Hamiltonian Path Graph is Connected for Simple s,t Paths in Rectangular Grid Graphs
[article]

2022
*
arXiv
*
pre-print

A simple s,t path P

arXiv:2205.08025v1
fatcat:akseq5h7tjbb5hgymx2gjxeuam
*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*...##
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Radial drawings of graphs: Geometric constraints and trade-offs

2008
*
Journal of Discrete Algorithms
*

The following requirements are considered: vertex centrality, edge crossings, curve

doi:10.1016/j.jda.2006.12.007
fatcat:rfk2r5l42nhavncw4cixt2c5au
*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*. ...##
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Embedding Planar Graphs at Fixed Vertex Locations
[chapter]

1998
*
Lecture Notes in Computer Science
*

Let G be a planar

doi:10.1007/3-540-37623-2_20
fatcat:hbycpmuhfbcbzkl3clkafpyrs4
*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. ...##
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Manhattan-Geodesic Embedding of Planar Graphs
[chapter]

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 ...

##
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Simultaneous Embedding of Planar Graphs
[article]

2015
*
arXiv
*
pre-print

This forms the basis for the visualization of dynamic

arXiv:1204.5853v3
fatcat:op5qpoub7bfhlgljxqygzx6jha
*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*. ...##
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Not being (super)thin or solid is hard: A study of grid Hamiltonicity

2009
*
Computational geometry
*

For many classes of

doi:10.1016/j.comgeo.2008.11.004
fatcat:is4zzl2opvgapi4tn6rovmsdlm
*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). ...##
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Vertex Intersection Graphs of Paths on a Grid

2012
*
Journal of Graph Algorithms and Applications
*

We investigate the class of vertex intersection

doi:10.7155/jgaa.00253
fatcat:txmeh3fdxfgghhx4dguptafhri
*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*. ...##
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The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes

2003
*
Theoretical Computer Science
*

Comput. 8 (1979) 410 -421) showed that the problem of computing the number of simple s-t paths

doi:10.1016/s0304-3975(03)00080-x
fatcat:vybthjqqabaapbpsdgeftyuedq
*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 ...
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