Results 1 to 10 of about 1,484,168 (305)

Untangling a Planar Graph [PDF]

Discrete & Computational Geometry, 2008
A straight-line drawing $δ$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,δ)$ denote the minimum number of vertices that need to be moved to untangle $δ$. We show that shift$(G,δ)$ is NP-hard to compute and to approximate.
Xavier Goaoc, Jan Kratochvíl, Yoshio Okamoto, Chan-Su Shin, Andreas Spillner 0001, Alexander Wolff 0001   +5 more
openaire   +4 more sources

The Odd Chromatic Number of a Planar Graph is at Most 8 [PDF]

Graphs and Combinatorics, 2022
Petruševski and Škrekovski recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph G is said to be odd if for each non-isolated vertex $$x \in V(G)$$ x ∈ V ( G ) there exists a colour c appearing an odd number ...
J. Petr, Julien Portier
semanticscholar   +1 more source

An improved planar graph product structure theorem [PDF]

Electronic Journal of Combinatorics, 2021
Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every planar graph $G$ there is a graph $H$ with treewidth at most 8 and a path $P$ such that $G\subseteq H\boxtimes P$. We improve this result by replacing "treewidth at most
T. Ueckerdt, David R. Wood, Wendy Yi
semanticscholar   +1 more source

Shallow Minors, Graph Products, and Beyond-Planar Graphs [PDF]

SIAM Journal on Discrete Mathematics, 2021
The planar graph product structure theorem of Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path.
Robert Hickingbotham, David R. Wood
semanticscholar   +1 more source

The structure and the list 3-dynamic coloring of outer-1-planar graphs [PDF]

Discrete Mathematics & Theoretical Computer Science, 2021
An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge.
Yan Li, Xin Zhang
doaj   +1 more source

Quantum approximate optimization of non-planar graph problems on a planar superconducting processor [PDF]

Nature Physics, 2020
Faster algorithms for combinatorial optimization could prove transformative for diverse areas such as logistics, finance and machine learning. Accordingly, the possibility of quantum enhanced optimization has driven much interest in quantum technologies.
F. Arute, K. Arya, R. Babbush, D. Bacon, J. Bardin, R. Barends, S. Boixo, M. Broughton, B. Buckley, D. Buell, B. Burkett, N. Bushnell, Yu Chen, Zijun Chen, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. Dunsworth, E. Farhi, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, S. Habegger, M. Harrigan, A. Ho, Sabrina Hong, Trent Huang, L. Ioffe, S. Isakov, E. Jeffrey, Zhang Jiang, Cody Jones, D. Kafri, K. Kechedzhi, J. Kelly, Seon Kim, P. Klimov, A. Korotkov, F. Kostritsa, D. Landhuis, P. Laptev, Mike Lindmark, M. Leib, E. Lucero, O. Martin, J. Martinis, J. McClean, M. McEwen, A. Megrant, X. Mi, M. Mohseni, W. Mruczkiewicz, J. Mutus, O. Naaman, M. Neeley, C. Neill, F. Neukart, H. Neven, M. Niu, T. O’Brien, B. O’Gorman, E. Ostby, A. Petukhov, Harald Putterman, C. Quintana, P. Roushan, N. Rubin, D. Sank, K. Satzinger, Andrea Skolik, V. Smelyanskiy, D. Strain, Michael Streif, Kevin J Sung, M. Szalay, A. Vainsencher, T. White, Z. Yao, P. Yeh, Adam Zalcman, Leo Zhou   +83 more
semanticscholar   +1 more source

Improved product structure for graphs on surfaces [PDF]

Discrete Mathematics & Theoretical Computer Science, 2022
Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes K_{\max\{2g,3\}}$. We improve
Marc Distel, Robert Hickingbotham, Tony Huynh, David R. Wood   +3 more
doaj   +1 more source

DiGress: Discrete Denoising diffusion for graph generation [PDF]

International Conference on Learning Representations, 2022
This work introduces DiGress, a discrete denoising diffusion model for generating graphs with categorical node and edge attributes. Our model utilizes a discrete diffusion process that progressively edits graphs with noise, through the process of adding ...
Clément Vignac, Igor Krawczuk, Antoine Siraudin, Bohan Wang, V. Cevher, P. Frossard   +5 more
semanticscholar   +1 more source

Planar Projections of Graphs [PDF]

Discrete Applied Mathematics, 2020
We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices has a representation in $\lceil \sqrt{n/2}+1 \rceil$ planes.
N. R. Aravind, Udit Maniyar
openaire   +4 more sources

On Optimal Beyond-Planar Graphs

Computing in Geometry and Topology, 2023
A graph is  beyond-planar if it can be drawn in the plane with a specific restriction on crossings. Several types of beyond-planar graphs have been investigated, such as k-planar graphs where every edge is crossed at most k times and RAC graphs where ...
Franz Brandenburg
doaj   +1 more source