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- W4294400211 abstract "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 number of graphs without $k+1$ disjoint minors in ${mathcal B}$ and (b) the properties of a uniformly random graph drawn from all such graphs on vertices ${1,dots,n}$. We present new results in the case when ${rm ex ,} {mathcal B}$ contains arbitrarily large fans for a general (good enough) set of forbidden minors ${mathcal B}$. A particular case where our results hold is ${mathcal B} = {K_4}$. For any fixed $k = 1, 2, dots$ we derive precise asymptotic counting formulas and describe 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 structure with $2k+1$ 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." @default.
- W4294400211 created "2022-09-03" @default.
- W4294400211 creator A5021094928 @default.
- W4294400211 date "2015-04-30" @default.
- W4294400211 modified "2023-09-30" @default.
- W4294400211 title "On graphs containing few disjoint excluded minors. Asymptotic number and structure of graphs containing few disjoint minors K4" @default.
- W4294400211 doi "https://doi.org/10.48550/arxiv.1504.08107" @default.
- W4294400211 hasPublicationYear "2015" @default.
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