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Forbidding Kuratowski Graphs as Immersions

2014
*
Journal of Graph Theory
*

In this note we give a structural characterization of the

doi:10.1002/jgt.21790
fatcat:lrmdqcrwzvc4bb5p4mdd3d4cz4
*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*...##
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Forbidding Kuratowski Graphs as Immersions
[article]

2012
*
arXiv
*
pre-print

In this note we give a structural characterization of the

arXiv:1207.5329v1
fatcat:xwnu4x2qt5fovjfirhghip4ari
*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*...##
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The structure of graphs with no K_3,3 immersion
[article]

2018
*
arXiv
*
pre-print

The

arXiv:1810.12873v2
fatcat:ojpwehrlxjdzdfmovkfr2p623u
*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. ...##
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Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing
[chapter]

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. ...

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

2011
*
arXiv
*
pre-print

A non-1-planar

arXiv:1110.4881v1
fatcat:cgluarxgq5h3zarnj7jtzewmzq
*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*. ...##
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A Linear Kernel for Planar Total Dominating Set
[article]

2017
*
arXiv
*
pre-print

This result complements several known constructive linear kernels on planar

arXiv:1211.0978v4
fatcat:kgsu6zq6b5gm7eobg43lnfa7qi
*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. ...##
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Helly groups
[article]

2021
*
arXiv
*
pre-print

Helly

arXiv:2002.06895v2
fatcat:b5pldtmlwfgb7budofbr66hhoi
*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. ...##
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Crux, space constraints and subdivisions
[article]

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. ...

##
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An interview with Stan Ulam and Mark Kac

1985
*
Journal of statistical physics
*

It would be interesting to plot a

doi:10.1007/bf01008345
fatcat:fcnjqyejjzd6tbukv5z63ekgzi
*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. ...##
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Chromatic and structural properties of sparse graph classes

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] ). ...

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

2013
unpublished

A

fatcat:niffvydnxjbgroiw2w6cejbhny
*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. ...##
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Proceedings of the 2021 Joint Workshop of the German Research Training Groups in Computer Science. May 31–June 1, 2021

2021

Such a result comes from Giannopoulou, Kaminski and Thilikos who gave a structural characterization for

doi:10.25593/opus4-fau-16426
fatcat:isoyq5lzdffxrdqzvkmfsnjgau
*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*. ...##
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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. ...