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Forbidding Kuratowski Graphs as Immersions

Archontia C. Giannopoulou, Marcin Kamiński, Dimitrios M. Thilikos
2014 Journal of Graph Theory  
In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions.  ...  Kuratowski graphs, namely K 5 and K 3,3 , give a precise characterization of planar graphs when excluded as topological minors.  ...  Introduction A famous graph theoretic result is the theorem of Kuratowski which states that a graph G is planar if and only if it does not contain K 5 and K 3,3 (also known as the Kuratowski graphs) as  ... 
doi:10.1002/jgt.21790 fatcat:lrmdqcrwzvc4bb5p4mdd3d4cz4

Forbidding Kuratowski Graphs as Immersions [article]

Archontia C. Giannopoulou, Marcin Kaminski, Dimitrios M. Thilikos
2012 arXiv   pre-print
In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions.  ...  Kuratowski graphs, namely K_5 and K_3,3, give a precise characterization of planar graphs when excluded as topological minors.  ...  Introduction A famous graph theoretic result is the theorem of Kuratowski which states that a graph G is planar if and only if it does not contain K 5 and K 3,3 (also known as the Kuratowski graphs) as  ... 
arXiv:1207.5329v1 fatcat:xwnu4x2qt5fovjfirhghip4ari

The structure of graphs with no K_3,3 immersion [article]

Matt DeVos, Mahdieh Malekian
2018 arXiv   pre-print
The Kuratowski-Wagner Theorem asserts that a graph is planar if and only if it does not have either K_3,3 or K_5 as a minor.  ...  In this paper we give a precise structure theorem for graphs which do not contain K_3,3 as an immersion.  ...  Accordingly, the outcome of this theorem remains unchanged if we forbid both K 3,3 and K 5 as immersions. 1.4.  ... 
arXiv:1810.12873v2 fatcat:ojpwehrlxjdzdfmovkfr2p623u

Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing [chapter]

Vladimir P. Korzhik, Bojan Mohar
2009 Lecture Notes in Computer Science  
As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný.  ...  A non-1-planar graph G is minimal if the graph G − e is 1-planar for every edge e of G.  ...  Hence, if a 1-planar graph G contains as a subgraph a PN-graph H, then in every plane 1-immersion of G the subgraph H is 1-immersed in the plane in the same way.  ... 
doi:10.1007/978-3-642-00219-9_29 fatcat:a2xzhcjf6fe5zmjcjkfbj232yq

Minimal obstructions for 1-immersions and hardness of 1-planarity testing [article]

Vladimir P. Korzhik, Bojan Mohar
2011 arXiv   pre-print
A non-1-planar graph G is minimal if the graph G-e is 1-planar for every edge e of G.  ...  We construct two infinite families of minimal non-1-planar graphs and show that for every integer n > 62, there are at least 2^(n-54)/4 nonisomorphic minimal non-1-planar graphs of order n.  ...  The authors are grateful to anonymous referee for pointing out that our proof in Section 5 might be used to derive a corresponding result for k-immersions.  ... 
arXiv:1110.4881v1 fatcat:cgluarxgq5h3zarnj7jtzewmzq

A Linear Kernel for Planar Total Dominating Set [article]

Valentin Garnero, Ignasi Sau
2017 arXiv   pre-print
This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating  ...  Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J.  ...  Forbidding Kuratowski graphs as immersions. Journal of Graph Theory, 78(1):43–60, 2015. 22. Q. Gu and N. Imani.  ... 
arXiv:1211.0978v4 fatcat:kgsu6zq6b5gm7eobg43lnfa7qi

Helly groups [article]

Jérémie Chalopin, Victor Chepoi, Anthony Genevois, Hiroshi Hirai, Damian Osajda
2021 arXiv   pre-print
Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs.  ...  In this article we focus on groups acting geometrically on Helly graphs – Helly groups.  ...  Hyperbolicity and Helly graphs. In Helly graphs, hyperbolicity can be characterized by forbidding isometric square-grids. Proposition 4.7.  ... 
arXiv:2002.06895v2 fatcat:b5pldtmlwfgb7budofbr66hhoi

Crux, space constraints and subdivisions [article]

Seonghyuk Im, Jaehoon Kim, Younjin Kim, Hong Liu
2022 arXiv   pre-print
As corollaries, we obtain (1) a characterisation of extremal graphs for which the square-root bound above is tight: they are essentially disjoint union of graphs each of which has the crux size linear  ...  in d; (2) a unifying approach to find a clique subdivision of almost optimal size in graphs which does not contain a fixed bipartite graph as a subgraph; (3) and that the clique subdivision size in random  ...  We would like to again emphasise that the only effect of forbidding a fixed bipartite graph is to relax the space constraints.  ... 
arXiv:2207.06653v1 fatcat:2xouxln6ffdadnqhtaib4756ja

An interview with Stan Ulam and Mark Kac

Mitchell Feigenbaum
1985 Journal of statistical physics  
It would be interesting to plot a graph: by the time it is five authors, the graph hits zero. ULAM: In mathematics it is zero. It is not uncommon in physics.  ...  People were so immersed in the actual problems. Occasionally there would be some kind of speculation about the more remote future.  ... 
doi:10.1007/bf01008345 fatcat:fcnjqyejjzd6tbukv5z63ekgzi

Chromatic and structural properties of sparse graph classes

Daniel Quiroz, Jan van den Heuvel
2017
Graphs are used to model many problems arising in areas such as physics, sociology, and computer science.  ...  Restriction can come from requiring graphs to be embeddable in a particular surface, to admit certain types of decompositions, or by forbidding some substructure.  ...  Two very famous results characterise planar graphs in terms of graph they cannot "contain". Theorem 1.1.1 (Kuratowski [56] ).  ... 
doi:10.21953/lse.yczhu2j6vv7n fatcat:ijrfxnc6ebdkzl2nu3u3ysed4e

Techniques combinatoires pour les algorithmes paramétrés et les noyaux, avec applications aux problèmes de multicoupe. Sciences et Techniques du Languedoc THÈSE

Jean Daligault
2013 unpublished
A graph has treewidth at most k if and only if it admits no bramble of order k + 1. Theorem 54 ( 54 Kuratowski). A graph is planar if and only if it does not have K 5 and K 3,3 as a minor.  ...  -Topological minor testing and immersion testing are FPT as well [76] . -Computing an optimal tree decomposition (parameterized by the treewidth) is FPT [9] .  ...  In general graphs, a request can be seen as the set of paths between its endpoints.  ... 
fatcat:niffvydnxjbgroiw2w6cejbhny

Proceedings of the 2021 Joint Workshop of the German Research Training Groups in Computer Science. May 31–June 1, 2021

Joint Workshop Of The German Research Training Groups In Computer Science, Erlangen May 31–June 1, 2021, Felix Freiling
2021
Such a result comes from Giannopoulou, Kaminski and Thilikos who gave a structural characterization for graphs forbidding Kuratowski graphs as immersions.  ...  One of the most popular graph classes characterized through forbidding the existence of both K 5 and K 3,3 as minors, are planar graphs.  ...  The task is to link the entity mentions in an unstructured text to their corresponding entities in a knowledge base/knowledge graph.  ... 
doi:10.25593/opus4-fau-16426 fatcat:isoyq5lzdffxrdqzvkmfsnjgau

Dagstuhl Reports, Volume 6, Issue 1, January 2016, Complete Issue [article]

2016
As a result, research fields are unnecessarily difficult for new researchers to enter, and their results challenging for outside communities to make use of.  ...  Research surveys and lecture notes are at least generally recognized by researchers as having some value, but writing for practitioners or a general audience is infrequently rewarded, and these latter  ...  Topological minor relation is a graph containment relation that is closely related to the minor and the immersion relations.  ... 
doi:10.4230/dagrep.6.1 fatcat:vq74ezrwifbkhex6twvtdzbpaa