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Graph Minors and Parameterized Algorithm Design [PDF]
The Multivariate Algorithmic Revolution and Beyond, 2012 The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory.
D. Thilikos
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Random Structures & Algorithms, 2005
AbstractWe study here lifts and random lifts of graphs, as defined by Amit and Linial (Combinatorica 22 (2002), 1–18). We consider the Hadwiger number η and the Hajós number σ of ℓ‐lifts of Kn and analyze their extremal as well as their typical values (that is, for random lifts). When ℓ = 2, we show that ${n \over 2} \leq \eta \leq n$, and random lifts
Yotam Drier, Nathan Linial
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AbstractWe study here lifts and random lifts of graphs, as defined by Amit and Linial (Combinatorica 22 (2002), 1–18). We consider the Hadwiger number η and the Hajós number σ of ℓ‐lifts of Kn and analyze their extremal as well as their typical values (that is, for random lifts). When ℓ = 2, we show that ${n \over 2} \leq \eta \leq n$, and random lifts
Yotam Drier, Nathan Linial
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Chi-boundedness of graph classes excluding wheel vertex-minors [PDF]
Journal of Combinatorial Theory Series B, 2019 A class of graphs is χ-bounded if there exists a function f:N→N such that for every graph G in the class and an induced subgraph H of G, if H has no clique of size q+1, then the chromatic number of H is less than or equal to f(q).
Hojin Choi, O-Joung Kwon, Sang-Il Oum
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Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds
International Colloquium on Automata, Languages and Programming, 2016Given a graph where vertices are partitioned into $k$ terminals and non-terminals, the goal is to compress the graph (i.e., reduce the number of non-terminals) using minor operations while preserving terminal distances approximately.The distortion of a ...
Yun Kuen Cheung +2 more
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Optimizing the Graph Minors Weak Structure Theorem
SIAM Journal on Discrete Mathematics, 2013 One of the major results of [N. Robertson and P. D. Seymour, Graph minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B, 63 (1995), pp. 65-110], also known as the weak structure theorem, reveals the local structure of graphs excluding some ...
Archontia C Giannopoulou +1 more
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Excluding Minors in Cubic Graphs
Combinatorics, Probability and Computing, 1996Let P10\e be the graph obtained by deleting an edge from the Petersen graph. We give a decomposition theorem for cubic graphs with no minor isomorphic to P10\e. The decomposition is used to show that graphs in this class are 3-edge-colourable. We also consider an application to a conjecture due to Grötzsch which states that a planar graph is 3-edge ...
Kyriakos Kilakos, F. Bruce Shepherd
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A Note on Oct1+-Minor-Free Graphs and Oct2+-Minor-Free Graphs
Journal of Interconnection Networks, 2022Let [Formula: see text] and [Formula: see text] be the planar and non-planar graphs, respectively, obtained from the Octahedron by 3-splitting a vertex. For [Formula: see text], we prove that if a 4-connected graph is [Formula: see text]-minor-free, then it is [Formula: see text], [Formula: see text] [Formula: see text] or it is obtained from [Formula:
Wenyan Jia +3 more
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Vertex-minors of graphs: A survey
Discrete Applied MathematicszbMATH Open Web Interface contents unavailable due to conflicting licenses.
Donggyu Kim, Sang-il Oum
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Minors in graphs of large girth
Random Structures & Algorithms, 2003AbstractWe show that for every odd integer g ≥ 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr(g+1)/4. This is best possible up to the value of the constant c for g = 5, 7, and 11.
Daniela Kühn, Deryk Osthus
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Towards the Graph Minor Theorems for Directed Graphs
2015Two key results of Robertson and Seymour's graph minor theory are:1.a structure theorem stating that all graphs excluding some fixed graph as a minor have a tree decomposition into pieces that are almost embeddable in a fixed surface.2.the k-disjoint paths problem is tractable when $$k$$ is a fixed constant: given a graph $$G$$ and $$k$$ pairs $$s_1 ...
Kawarabayashi, Ken-Ichi +1 more
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