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Labeled K2, t Minors in Plane Graphs

Thomas Böhme, Bojan Mohar
2002 Journal of combinatorial theory. Series B (Print)  
It is shown that G contains a K 2, 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.  ... 
doi:10.1006/jctb.2001.2083 fatcat:tlki2yzit5e3fnhwaedn75mwl4

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.  ... 

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 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.”  ... 

Universality in minor-closed graph classes [article]

Tony Huynh and Bojan Mohar and Robert Šámal and Carsten Thomassen and David R. Wood
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.  ... 
arXiv:2109.00327v1 fatcat:wmv5mwtfvjfttfhjmxuswpn27e

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  ... 

Refined Vertex Sparsifiers of Planar Graphs [article]

Robert Krauthgamer, Havana Rika
2019 arXiv   pre-print
Our third contribution is a duality between cut sparsification and distance sparsification for certain planar 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.  ... 
arXiv:1702.05951v3 fatcat:e3ot4ry5sbg35nbvqt3qqbwpae

On obstructions to small face covers in planar graphs

D Bienstock, N Dean
1992 Journal of combinatorial theory. Series B (Print)  
If the embedding of the 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).  ... 
doi:10.1016/0095-8956(92)90040-5 fatcat:6qnv2u7klzgs5n5ldt326gqdke

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.  ... 

Combinatorial identities by way of Wilf's multigraph model

Theresa L. Friedman, Paul Klingsberg
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 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 ≥ tt 0 }.  ... 
doi:10.1155/ijmms/2006/96327 fatcat:dpmcs63isfexvm2bmu2yo6tp5u

Planar graphs with the maximum number of induced 4-cycles or 5-cycles [article]

Michael Savery
2021 arXiv   pre-print
We show that K_2,n-2 uniquely achieves this maximum 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.  ... 
arXiv:2108.00526v2 fatcat:j42nfi3fj5dfxa3j22dqt2a4z4

Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms

Guido Gentile, Michele V. Bartuccelli, Jonathan H. B. Deane
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.  ... 
doi:10.1063/1.1926208 fatcat:h7h6v4h3xvhw7ntgy3cbfiqiie

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

Farshid Arman, J. K. Aggarwal
1993 ACM Computing Surveys  
A comprehensive survey of the recent publications 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.  ... 
doi:10.1145/151254.151255 fatcat:4oblkspfz5hjdiug6zegiw6y7y

Around matrix-tree theorem

Yuri Burman, Boris Shapiro
2006 Mathematical Research Letters  
Generalizing the classical matrix-tree theorem we provide a formula counting, for a given 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  ... 
doi:10.4310/mrl.2006.v13.n5.a7 fatcat:75jx2fwwyranpdwd7b3ozetkrm

Forbidden minors to graphs with small feedback sets

Michael J. Dinneen, Kevin Cattell, Michael R. Fellows
2001 Discrete Mathematics  
Finite obstruction set characterizations for lower ideals 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] .  ... 
doi:10.1016/s0012-365x(00)00083-2 fatcat:altsqcza6ne4vgm4r7wurilv24

Around matrix-tree theorem [article]

Yurii Burman, Boris Shapiro
2006 arXiv   pre-print
Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given 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).  ... 
arXiv:math/0512164v2 fatcat:2u5ep3lrljfmvf4iz2z6bg62fm
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