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2-Layer k-Planar Graphs: Density, Crossing Lemma, Relationships, and Pathwidth
[article]
2020
arXiv
pre-print
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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 . ...
arXiv:2008.09329v1
fatcat:q3hyggamubcjvgxecznztl7bdu
Book Embeddings of Graph Products
[article]
2020
arXiv
pre-print
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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. ...
arXiv:2007.15102v1
fatcat:hpjrwfbyvvbtbg54hbes3wdbhe
NIC-Planar Graphs
[article]
2017
arXiv
pre-print
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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. ...
arXiv:1701.04375v3
fatcat:gowtlgenuzh4hpztqkd4pnfiha
Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seaminar 19092)
2019
Dagstuhl Reports
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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. ...
doi:10.4230/dagrep.9.2.123
dblp:journals/dagstuhl-reports/Hong0PT19
fatcat:pvtbranvp5fwfkaul2nlmb6m4e
Approximate MAP Estimation for Pairwise Potentials via Baker's Technique
[article]
2018
arXiv
pre-print
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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. ...
arXiv:1412.0340v4
fatcat:bwu3jpcq3fc5bgpugzpvmfwnri
Approximation metatheorems for classes with bounded expansion
[article]
2021
arXiv
pre-print
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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] ...
arXiv:2103.08698v3
fatcat:yre46davf5eadmdzews752djle
Anagram-free Graph Colouring and Colour Schemes
2019
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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. ...
doi:10.26180/5c72eca26d5c7
fatcat:i3g3beevpvc67osu2qyeepwabm
Cover graphs and order dimension
[article]
2017
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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. ...
doi:10.14279/depositonce-5807
fatcat:h3klm4akevfptn42cvmnkwctdq
Minimal matrices in the Bruhat order for symmetric (0,1)-matrices
2017
Linear Algebra and its Applications
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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. ...
doi:10.1016/j.laa.2017.05.014
fatcat:zlcp5rb2sjeqvlyaajvf2nyoby
Fixed-Parameter Tractability and Completeness I: Basic Results
1995
SIAM journal on computing (Print)
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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. ...
doi:10.1137/s0097539792228228
fatcat:lrps5o3r3jg7jcy7ellmmdytoa
Dagstuhl Reports, Volume 9, Issue 2, February 2019, Complete Issue
[article]
2019
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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. ...
doi:10.4230/dagrep.9.2
fatcat:ldqjjo5eqffszji56jhkmocl34
Approximation algorithms for network design and cut problems in bounded-treewidth
[article]
2021
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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). ...
doi:10.22028/d291-32983
fatcat:v6rutlrvo5bafgggqawirxdxhm
LIPIcs, Volume 135, TQC'19, Complete Volume
[article]
2019
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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). ...
doi:10.4230/lipics.tqc.2019
fatcat:36nxd6j4fzeyxppe6c5osytd3q
Dagstuhl Reports, Volume 4, Issue 11, November 2014, Complete Issue
[article]
2015
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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). ...
doi:10.4230/dagrep.4.11
fatcat:zzps76cx7rfldla5z5dikvsqbe
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)
unpublished
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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) . ...
fatcat:iz3co5xisfejfbnx2t6ptrorfm
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