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### 2-Layer k-Planar Graphs: Density, Crossing Lemma, Relationships, and Pathwidth [article]

Patrizio Angelini, Giordano Da Lozzo, Henry Förster, Thomas Schneck
2020 arXiv   pre-print
Finally, we study relationships between k-planarity and h-quasiplanarity in the 2-layer model and show that 2-layer k-planar graphs have pathwidth at most k+1.  ...  Based on these results, we provide a Crossing Lemma for 2-layer k-planar graphs, which then implies a general density bound for 2-layer k-planar graphs.  ...  Every 2-layer k-planar graph has pathwidth at most k + 1. Proof. Let G be a topological 2-layer k-planar graph with parts U and V .  ...

### Book Embeddings of Graph Products [article]

Sergey Pupyrev
2020 arXiv   pre-print
A k-stack layout (also called a k-page book embedding) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing edges with respect to the vertex order  ...  We show that the stack number is bounded for the strong product of a path and (i) a graph of bounded pathwidth or (ii) a bipartite graph of bounded treewidth and bounded degree.  ...  Bekos et al. proved that 1-planar graphs have bounded stack number [3] . Recall that a graph is k-planar if it can be drawn in the plane with at most k crossings per edge.  ...

### NIC-Planar Graphs [article]

Christian Bachmaier, Franz J. Brandenburg, Kathrin Hanauer, Daniel Neuwirth, Josef Reislhuber
2017 arXiv   pre-print
The characterization is used to derive tight bounds on the density of maximal NIC-planar graphs which ranges between 3.2(n-2) and 3.6(n-2).  ...  A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex.  ...  NIC-planar graphs were introduced by Zhang [39] , who proved a density of at most 3.6(n − 2) and showed that K 6 is not NIC-planar.  ...

### Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seaminar 19092)

Seok-Hee Hong, Michael Kaufmann, János Pach, Csaba D. Tóth, Michael Wagner
2019 Dagstuhl Reports
algorithms for so-called beyond-planar graphs, i.e., non-planar graphs that admit a drawing with topological constraints such as specific types of crossings, or with some forbidden crossing patterns.  ...  This report documents the program and the outcomes of Dagstuhl Seminar 19092 "Beyond-Planar Graphs: Combinatorics, Models and Algorithms" which brought together 36 researchers in the areas of graph theory  ...  We thank Günter Rote and Martin Gronemann for asking interesting questions that led to some of this research.  ...

### Approximate MAP Estimation for Pairwise Potentials via Baker's Technique [article]

Yi-Kai Wang
2018 arXiv   pre-print
We show that computing MAX 2-CSP admits polynomial-time approximation scheme (PTAS) on planar graphs, graphs with bounded local treewidth, H-minor-free graphs, geometric graphs with bounded density and  ...  This implies computing MAX-CUT, MAX-DICUT and MAX k-CUT admits PTASs on all these classes of graphs.  ...  By a similar argument, we achieve the 1 − ∆+1 k+2 = k k+2 optimal solution in O(q O(k) kn) time. Bounded-crossing-number Graphs For bounded-crossing-number graphs, ∆ = ϕ + 1.  ...

### Approximation metatheorems for classes with bounded expansion [article]

Zdeněk Dvořák
2021 arXiv   pre-print
We obtain * constant-factor approximation algorithm in any class of graphs with bounded expansion, * a QPTAS in any class with strongly sublinear separators, and * a PTAS in any fractionally treewidth-fragile  ...  . • Baker [3] designed a very powerful technique for planar graphs based on finding a partition of the graph into layers (where the edges are al- lowed only within the layers and between consecutive layers  ...  This includes • planar graphs [27] , and more generally all proper minor-closed classes [2] ; • graphs drawn in the plane (or on a fixed surface) with a bounded number of crossings on each edge [34]  ...

### Anagram-free Graph Colouring and Colour Schemes

TIMOTHY EDWARD WILSON
2019
Graphs model the connectivity of networks, and many researchers study how to colour the nodes of a graph under various restrictions.  ...  Anagram-free colouring is a type of graph colouring which requires the first half of every path through a graph to not be a permutation of the second half.  ...  Acknowledgements Most of all, I must thank David Wood, who has been an excellent supervisor and a rich source of advice and collaboration.  ...

### Cover graphs and order dimension [article]

Veit Wiechert, Technische Universität Berlin, Technische Universität Berlin, Stefan Felsner
2017
The goal of this dissertation is to study the various connections between the dimension of posets and graph theoretic properties of their cover graphs.  ...  Requiring cover graphs to be planar though is not enough to guarantee an upper bound on the dimension, as is witnessed by the famous construction of Kelly.  ...  Consider the graphs for which there exists an embedding in the plane such that each edge is crossed at most k times. Those graphs are called k-planar graphs.  ...

### Minimal matrices in the Bruhat order for symmetric (0,1)-matrices

Henrique F. da Cruz, Rosário Fernandes, Susana Furtado
2017 Linear Algebra and its Applications
graph theory and combinatorics.  ...  We have received 150 abstracts for 20-minute contributed talks which will run in four parallel sessions and cover a large variety of areas within graph theory and combinatorics.  ...  We show that, unlike for trees, all containments in the Venn diagram are proper for MOPs, a subfamily of 2-trees.  ...

### Fixed-Parameter Tractability and Completeness I: Basic Results

Rod G. Downey, Michael R. Fellows
1995 SIAM journal on computing (Print)
DOMINATING SET is shown to be complete for W[2], and thus is not fixed-parameter tractable unless INDEPENDENT SET, CLIQUE, IRREDUNDANT SET, and many other natural problems in W[2] are also fixed-parameter  ...  In particular, we define a hierarchy of classes of parameterized problems FPT _ W[I] c_ W[2] _ _ _ _ _ _ W[SAT] c_ W[P] and identify natural complete problems for W[t] for > 2.  ...  The general version of this problem is P SP AC E-complete by a reduction from QBF (see Garey and Johnson [81, GP1] [2] ). Acknowledgments.  ...

### Dagstuhl Reports, Volume 9, Issue 2, February 2019, Complete Issue [article]

2019
Dagstuhl Reports, Volume 9, Issue 2, February 2019, Complete Issue  ...  We plan to identify, organise and share the different sources of funding that could be used for AI for Social Good projects. We listed possible future meetings and workshops.  ...  To that end, we defined multiple milestones and commitments, we built working groups, etc. Acknowledgments.  ...

### Approximation algorithms for network design and cut problems in bounded-treewidth [article]

Daniel Vaz, Universität Des Saarlandes
2021
We show an O(log 2 n)-approximation algorithm for boundedtreewidth graphs, matching the known lower bound for trees, and improving the best possible result using previous techniques.  ...  We study the approximability of these problems on trees and bounded-treewidth graphs.  ...  A planar graph is a graph that can be embedded onto the plane (R 2 ) in such a way that edges only intersect at their endpoints (i.e. there are no crossings).  ...

### LIPIcs, Volume 135, TQC'19, Complete Volume [article]

Wim van Dam, Laura Mančinska
2019
Acknowledgements The authors would like to thank Aniruddha Bapat for insights on the hierarchical product of graphs and suggestions for tightening the routing lower bound for these graphs.  ...  Acknowledgements We thank Steven Herbert for many helpful conversations and encouragement. 1 Acknowledgements We thank Chris Cade and Stacey Jeffery for helpful discussions. 1 Acknowledgements DG thanks  ...  We first show the following relationship between effective resistance and circuit rank: Lemma 9. Let r be the circuit rank of G(x).  ...

### Dagstuhl Reports, Volume 4, Issue 11, November 2014, Complete Issue [article]

2015
Firstly, we look into problems that are solvable in slightly-superexponential running time, i.e., O ú (2 O(k log k) ), and then we examine geometric and planar problems for which XP algorithms with running  ...  We are unaware of any natural graph parameters between pathwidth and treedepth and so our results provide a dichotomy of the complexity of MCPP.  ...  Next, we turn to algorithmic engineering aspects of single-source shortest-paths graph search (such as node labels, a joint graph and heap node representation and the proper use of C++-templates).  ...

### Reinforcement Learning (Dagstuhl Seminar 13321) The Critical Internet Infrastructure (Dagstuhl Seminar 13322) Coding Theory (Dagstuhl Seminar 13351) Interaction with Information for Visual Reasoning (Dagstuhl Seminar 13352)

Peter Auer, Marcus Hutter, Laurent, Georg Carle, Jochen Schiller, Steve Uhlig, Walter Willinger, Matthias Wählisch, Thore Husfeldt, Ramamohan Paturi, Gregory Sorkin, Ryan Williams (+14 others)
unpublished
Marc: Sensors are underrated, ON and OFF rates need to be studied. Biological OFF rate can be studied, OFF rate is the time for A and B to stay attached. Then, they dissociate.  ...  Should attach long enough and go to the membrane for reaction. David: There are indeed grand challenges in the IEEE area. Yet, we need to be cautious.  ...  Williams (STOC 2009) showed how to count simple paths on k vertices and matchings on k/2 edges in an n-vertex graph in time n k/2+O(1) .  ...
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