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Labeled K2, t Minors in Plane Graphs

2002
*
Journal of combinatorial theory. Series B (Print)
*

It is shown that G contains a K 2,

doi:10.1006/jctb.2001.2083
fatcat:tlki2yzit5e3fnhwaedn75mwl4
*t**minor*such that*t*is large and each vertex of degree 2*in*K 2,*t*corresponds to some vertex of U if and only if there is no small face cover of U. ... Let G be a 3-connected planar*graph*and let U ı V(G). ...*In*particular, G contains a U-*labeled*K 2,*t*-*minor*where*t*\ f(y(U)). Conversely, if G contains a U-*labeled*K 2,*t*-*minor*, then y(U) \*t*/2. ...##
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Page 5142 of Mathematical Reviews Vol. , Issue 97H
[page]

1997
*
Mathematical Reviews
*

We find a nonplanar

*graph*obtained from K3,3 which has a planar Kronecker product by*K2*. ... Concerning the mi- nor of a*graph*, we prove that for every connected*graph*G, G is a*minor*of GA K3, a*minor*of GAH where H has an odd cycle, and then a*minor*of GAG. ...##
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Page 1954 of Mathematical Reviews Vol. , Issue 95d
[page]

1995
*
Mathematical Reviews
*

The author shows that the reconstruction conjecture is true for planar

*graphs*with minimum degree 5 which can be embedded*in*the*plane**in*such a way that any vertex of degree 5.is incident with triangular ... Using the notion of covering, we prove that a*minor*-closed class of*graphs*can- not be recognized by local computations, except*in*a few special cases.” ...##
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Universality in minor-closed graph classes
[article]

2021
*
arXiv
*
pre-print

*graph*K_t with multiplicity

*t*, for every finite

*t*. ... More generally, for every positive integer

*t*we construct a countable

*graph*that contains every countable K_t-

*minor*-free

*graph*and has the above key properties. ... A

*plane*

*graph*is a

*graph*embedded

*in*the

*plane*, i.e. drawn without crossings. ...

##
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Page 3585 of Mathematical Reviews Vol. , Issue 92g
[page]

1992
*
Mathematical Reviews
*

Let s; denote the number of games won by player i

*in*a round- robin tournament*T*,,. ... A proper edge-colouring of a*graph*G is said to be standard if there exists an abelian group (A,+) for which the vertices of G can be*labelled*with elements of A*in*such a way that the colour of each edge ...##
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Refined Vertex Sparsifiers of Planar Graphs
[article]

2019
*
arXiv
*
pre-print

Our third contribution is a duality between cut sparsification and distance sparsification for certain planar

arXiv:1702.05951v3
fatcat:e3ot4ry5sbg35nbvqt3qqbwpae
*graphs*, when the sparsifier H is required to be a*minor*of G. ... Our second and main contribution is to refine the known bounds*in*terms of γ=γ(G), which is defined as the minimum number of faces that are incident to all the terminals*in*a planar*graph*G. ... Every planar network G admits a*minor*mimicking network of size O(*k2*2k ). Proof. ...##
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On obstructions to small face covers in planar graphs

1992
*
Journal of combinatorial theory. Series B (Print)
*

If the embedding of the

doi:10.1016/0095-8956(92)90040-5
fatcat:6qnv2u7klzgs5n5ldt326gqdke
*graph*is fixed, this problem leads to variants of the ErdGs-Posa theorem on independent cycles*in*a*graph*. ... Several algorithmic and*graph*-theoretic developments have focused on the problem of covering,*in*a planar*graph*, selected vertices with fewest possible faces. ... However, it turns out that*T*(2)\F(2) # (a-the*graph**in*Fig. 5 (a) is a member of*T*(2), but it contains as a proper*minor*the*graph**in*Fig. 5(b) , which is a member of F(2). ...##
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Page 6536 of Mathematical Reviews Vol. , Issue 93m
[page]

1993
*
Mathematical Reviews
*

*In*our discussions, the class of almost bipartite

*graphs*is defined and we show that if G is an almost bipartite

*graph*, then it is a

*minor*of Gx

*K2*. We conjecture that this is true for all

*graphs*.” ...

*In*this case the leaves are

*labeled*with (aligned) nucleic acid or protein sequences of extant taxa. To assess a result, it is important to have information about the distribution of weights. ...

##
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Combinatorial identities by way of Wilf's multigraph model

2006
*
International Journal of Mathematics and Mathematical Sciences
*

For many families of combinatorial objects, a construction of Wilf (1977) allows the members of the family to be viewed as paths

doi:10.1155/ijmms/2006/96327
fatcat:dpmcs63isfexvm2bmu2yo6tp5u
*in*a directed multigraph. ... ,*t*) = 2≤k1<*k2*<···<kt−1≤n−2 2*k2*−k1 3 k3−*k2*···(*t*− 1) kt−1−kt−2*t*(n−1)−kt−1 . (4.8) The well-known Prüfer bijection between*labeled*trees and sequences (e.g., described*in*[4] ) induces a bijection between ... A binomial family is a combinatorial family*graph*for which the following hold. (a) For some*t*0 ≥ 0, V is a set of lattice points*in*the*plane*, namely, V = {(n,*t*) : n ≥*t*≥*t*0 }. ...##
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Planar graphs with the maximum number of induced 4-cycles or 5-cycles
[article]

2021
*
arXiv
*
pre-print

We show that K_2,n-2 uniquely achieves this maximum

arXiv:2108.00526v2
fatcat:j42nfi3fj5dfxa3j22dqt2a4z4
*in*the C_4 case, and we identify the*graphs*which achieve the maximum*in*the C_5 case. ... For large n we determine exactly the maximum numbers of induced C_4 and C_5 subgraphs that a planar*graph*on n vertices can contain. ... Let n be large and let G be an n-vertex*plane**graph*. ...##
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Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms

2005
*
Journal of Mathematical Physics
*

*In*the case of quasi-periodic forcing terms we need Renormalization Group techniques

*in*order to control the small divisors arising

*in*the perturbation series. ...

*In*the limit of large damping we look for quasi-periodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties

*in*the inverse of the damping coefficient ...

*In*the last

*graph*one has the constraints k1 +

*k2*= k − 1 and ν1 + ν2 = ν. ...

##
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Model-based object recognition in dense-range images---a review

1993
*
ACM Computing Surveys
*

A comprehensive survey of the recent publications

doi:10.1145/151254.151255
fatcat:4oblkspfz5hjdiug6zegiw6y7y
*in*each subtask pertaining to dense-range image object recognition is presented. ... The goal*in*computer vision systems is to analyze data collected from the environment and derive an interpretation to complete a specified task. ... The principal directions are the roots of H T2 -*T*1 det E F G = 0, (A.1O) L MN where*T*= du/dv = tan 6, and 6 is the angle*in*the parameter (u, v)-*plane*. ...##
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Around matrix-tree theorem

2006
*
Mathematical Research Letters
*

Generalizing the classical matrix-tree theorem we provide a formula counting, for a given

doi:10.4310/mrl.2006.v13.n5.a7
fatcat:75jx2fwwyranpdwd7b3ozetkrm
*graph*, its subgraphs with a fixed 2-core. ... We are grateful to Professor Olivier Bernardi who pointed out an important mistake*in*an earlier version of this paper. ... Acknowledgments The first named author is sincerely grateful to the Mathematics Department of Stockholm University for the hospitality and financial support of his visit*in*September 2005 when the essential ...##
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Forbidden minors to graphs with small feedback sets

2001
*
Discrete Mathematics
*

Finite obstruction set characterizations for lower ideals

doi:10.1016/s0012-365x(00)00083-2
fatcat:altsqcza6ne4vgm4r7wurilv24
*in*the*minor*order are guaranteed to exist by the*graph**minor*theorem. ...*In*this paper we characterize several families of*graphs*with small feedback sets, namely k1-FEEDBACK VERTEX SET,*k2*-FEEDBACK EDGE SET and (k1;*k2*)-FEEDBACK VERTEX=EDGE SET, for small integer parameters ... The number of obstructions for embedding*in*the projective*plane*is 35 [19] . ...##
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Around matrix-tree theorem
[article]

2006
*
arXiv
*
pre-print

Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given

arXiv:math/0512164v2
fatcat:2u5ep3lrljfmvf4iz2z6bg62fm
*graph*with a fixed 2-core. ... The first named author is sincerely grateful to the Mathematics Department of Stockholm University for the hospitality and financial support of his visit*in*September 2005 when the essential part of this ... the Laplacian matrix (*in*the case of trees it was its principal*minor*). ...
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