Polynomial bounds for the Graph Minor Structure Theorem

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10515376" target="_blank" >RIV/00216208:11320/25:10515376 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1109/FOCS63196.2025.00104" target="_blank" >https://doi.org/10.1109/FOCS63196.2025.00104</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1109/FOCS63196.2025.00104" target="_blank" >10.1109/FOCS63196.2025.00104</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Polynomial bounds for the Graph Minor Structure Theorem

Popis výsledku v původním jazyce

The Graph Minor Structure Theorem, originally proven by Robertson and Seymour [JCTB, 2003], asserts that there exist functions f1, f2:N to N such that for every non-planar graph H with t := |V (H)|, every H-minor-free graph can be obtained via the clique-sum operation from graphs which embed into surfaces where H does not embed after deleting at most f&lt;inf&gt;1&lt;/inf&gt;(t) many vertices with up to at most t&lt;sup&gt;2&lt;/sup&gt; - 1 many &quot;vortices&quot;which are of &quot;depth&quot;at most f&lt;inf&gt;2&lt;/inf&gt;(t). In the proof presented by Robertson and Seymour the functions f&lt;inf&gt;1&lt;/inf&gt; and f&lt;inf&gt;2&lt;/inf&gt; are non-constructive. Kawarabayashi, Thomas, and Wollan [arXiv, 2020] found a new proof showing that f&lt;inf&gt;1&lt;/inf&gt;(t),f&lt;inf&gt;2&lt;/inf&gt;(t) Element 2&lt;sup&gt;poly(t)&lt;/sup&gt;. While believing that this bound was the best their methods could achieve, Kawarabayashi, Thomas, and Wollan conjectured that f&lt;inf&gt;1&lt;/inf&gt; and f&lt;inf&gt;2&lt;/inf&gt; can be improved to be polynomials.In this paper we confirm their conjecture and prove that f&lt;inf&gt;1&lt;/inf&gt;(t),f&lt;inf&gt;2&lt;/inf&gt;(t) Element O(t&lt;sup&gt;2300&lt;/sup&gt;). Our proofs are fully constructive and yield a polynomial-time algorithm that either finds H as a minor in a graph G or produces a clique-sum decomposition for G as above.

Název v anglickém jazyce

Polynomial bounds for the Graph Minor Structure Theorem

Popis výsledku anglicky

The Graph Minor Structure Theorem, originally proven by Robertson and Seymour [JCTB, 2003], asserts that there exist functions f1, f2:N to N such that for every non-planar graph H with t := |V (H)|, every H-minor-free graph can be obtained via the clique-sum operation from graphs which embed into surfaces where H does not embed after deleting at most f&lt;inf&gt;1&lt;/inf&gt;(t) many vertices with up to at most t&lt;sup&gt;2&lt;/sup&gt; - 1 many &quot;vortices&quot;which are of &quot;depth&quot;at most f&lt;inf&gt;2&lt;/inf&gt;(t). In the proof presented by Robertson and Seymour the functions f&lt;inf&gt;1&lt;/inf&gt; and f&lt;inf&gt;2&lt;/inf&gt; are non-constructive. Kawarabayashi, Thomas, and Wollan [arXiv, 2020] found a new proof showing that f&lt;inf&gt;1&lt;/inf&gt;(t),f&lt;inf&gt;2&lt;/inf&gt;(t) Element 2&lt;sup&gt;poly(t)&lt;/sup&gt;. While believing that this bound was the best their methods could achieve, Kawarabayashi, Thomas, and Wollan conjectured that f&lt;inf&gt;1&lt;/inf&gt; and f&lt;inf&gt;2&lt;/inf&gt; can be improved to be polynomials.In this paper we confirm their conjecture and prove that f&lt;inf&gt;1&lt;/inf&gt;(t),f&lt;inf&gt;2&lt;/inf&gt;(t) Element O(t&lt;sup&gt;2300&lt;/sup&gt;). Our proofs are fully constructive and yield a polynomial-time algorithm that either finds H as a minor in a graph G or produces a clique-sum decomposition for G as above.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/LL2328" target="_blank" >LL2328: Zobecnění věty o čtyřech barvách</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

Proceedings Annual IEEE Symposium on Foundations of Computer Science Focs

ISBN

979-8-3315-7132-0

ISSN

—

e-ISSN

—

Počet stran výsledku

18

Strana od-do

1961-1978

Název nakladatele

IEEE

Místo vydání

NEUVEDENO

Místo konání akce

Sydney, Australia

Datum konání akce

14. 12. 2025

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

001711633100097