English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let ${\rm ex \,} {\mathcal B}$ be a minor-closed class of graphs with a set${\mathcal B}$ of minimal excluded minors. We study (a) the asymptotic numberof graphs without $k+1$ disjoint minors in ${\mathcal B}$ and (b) theproperties of a uniformly random graph drawn from all such graphs on vertices$\{1,\dots,n\}$. We present new results in the case when ${\rm ex \,} {\mathcalB}$ contains arbitrarily large fans for a general (good enough) set offorbidden minors ${\mathcal B}$.  A particular case where our results hold is ${\mathcal B} = \{K_4\}$. For anyfixed $k = 1, 2, \dots$ we derive precise asymptotic counting formulas anddescribe the structure of typical graphs that have at most $k$ disjoint minors$K_4$. For $k = 0$ this is the well-known class of series-parallel graphs. For$k \ge 1$ we show that typical instances have an elaborate tree-like structurewith $2k+1$ special vertices of very high degree.  The proofs combine a variety of methods, including new structural results,Robertson and Seymour's graph minor theory and analytic combinatorics.

On graphs containing few disjoint excluded minors. Asymptotic number and structure of graphs containing few disjoint minors K4