A Constructive Classification of Graphs

M. A. Iordanskii

doi:10.18255/1818-1015-2012-4-144-153

A Constructive Classification of Graphs

Modeling and Analysis of Information Systems ◽

10.18255/1818-1015-2012-4-144-153 ◽

2015 ◽

Vol 19 (4) ◽

pp. 144-153

Author(s):

M. A. Iordanskii

Keyword(s):

Closed Graph ◽

Graph Classes ◽

Closed Class ◽

Classification Of Graphs

The classes of graphs closed regarding the set-theoretical operations of union and intersection are considered. Some constructive descriptions of the closed graph classes are set by the element and operational generating basses. Such bases have been constructed for many classes of graphs. The backward problems (when the generating bases are given and it is necessary to define the characteristic properties of corresponding graphs) are solved in the paper. Subsets of element and operational bases of the closed class of all graphs are considered as generating bases.

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Asymptotic Properties of Some Minor-Closed Classes of Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548314000303 ◽

2014 ◽

Vol 23 (5) ◽

pp. 749-795 ◽

Cited By ~ 1

Author(s):

MIREILLE BOUSQUET-MÉLOU ◽

KERSTIN WELLER

Keyword(s):

Asymptotic Properties ◽

Systematic Investigation ◽

Exact Enumeration ◽

Limit Laws ◽

Closed Class ◽

Connected Case ◽

Non Gaussian ◽

A Minor ◽

Excluded Minors

Let${\cal A}$be a minor-closed class of labelled graphs, and let${\cal G}_{n}$be a random graph sampled uniformly from the set ofn-vertex graphs of${\cal A}$. Whennis large, what is the probability that${\cal G}_{n}$is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected.Using exact enumeration, we study a collection of classes${\cal A}$excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating functionC(z) that counts connected graphs of${\cal A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-Gaussian limit laws (Beta and Gamma), and clearly merits a systematic investigation.

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A Faster Shortest-Paths Algorithm for Minor-Closed Graph Classes

Graph-Theoretic Concepts in Computer Science - Lecture Notes in Computer Science ◽

10.1007/978-3-540-92248-3_32 ◽

2008 ◽

pp. 360-371 ◽

Cited By ~ 1

Author(s):

Siamak Tazari ◽

Matthias Müller-Hannemann

Keyword(s):

Shortest Paths ◽

Closed Graph ◽

Graph Classes

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Complexity classification of the edge coloring problem for a family of graph classes

Discrete Mathematics and Applications ◽

10.1515/dma-2017-0011 ◽

2017 ◽

Vol 27 (2) ◽

Cited By ~ 1

Author(s):

Dmitriy S. Malyshev

Keyword(s):

Edge Coloring ◽

Complete Classification ◽

Chromatic Index ◽

Forbidden Subgraphs ◽

Coloring Problem ◽

Graph Classes ◽

Complexity Classification ◽

Index Problem

AbstractA class of graphs is called monotone if it is closed under deletion of vertices and edges. Any such class may be defined in terms of forbidden subgraphs. The chromatic index of a graph is the smallest number of colors required for its edge-coloring such that any two adjacent edges have different colors. We obtain a complete classification of the complexity of the chromatic index problem for all monotone classes defined in terms of forbidden subgraphs having at most 6 edges or at most 7 vertices.

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Smaller Extended Formulations for Spanning Tree Polytopes in Minor-closed Classes and Beyond

The Electronic Journal of Combinatorics ◽

10.37236/10522 ◽

2021 ◽

Vol 28 (4) ◽

Author(s):

Manuel Aprile ◽

Samuel Fiorini ◽

Tony Huynh ◽

Gwenaël Joret ◽

David R. Wood

Keyword(s):

Spanning Tree ◽

Short Proof ◽

Induced Subgraphs ◽

Direct Construction ◽

Extension Complexity ◽

Graph Classes ◽

Closed Class ◽

Vertex Graph ◽

Graph Class ◽

Strongly Sublinear

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$ have balanced separators of size $O(n^\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$ is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).

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