A polynomial excluded-minor approximation of treedepth

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a classof labelled graphs has smooth growth; and, for the random graphRnsampled uniformly from then-vertex graphs in, the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, asn→ ∞ the probability thatRnis connected tends to 1/A(ρ), whereA(x) is the exponential generating function forand ρ is its radius of convergence.

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An excluded minor theorem for the Octahedron plus an edge

Journal of Graph Theory ◽

10.1002/jgt.20272 ◽

2007 ◽

Vol 57 (2) ◽

pp. 124-130 ◽

Cited By ~ 2

Author(s):

John Maharry

Keyword(s):

Excluded Minor

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A Characterization of Almost-Planar Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548300002005 ◽

1996 ◽

Vol 5 (3) ◽

pp. 227-245 ◽

Cited By ~ 4

Author(s):

Bradley S. Gubser

Keyword(s):

Graph Theory ◽

Planar Graph ◽

Planar Graphs ◽

Explicit Description ◽

Famous Result ◽

Minor Criterion ◽

Excluded Minor

Kuratowski's Theorem, perhaps the most famous result in graph theory, states that K5 and K3,3 are the only non-planar graphs for which both G\e, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the almost-planar graphs, those non-planar graphs for which G\e or G/e is planar for all edges e of G. This paper gives two characterizations of the almost-planar graphs: an explicit description of the structure of almost-planar graphs; and an excluded minor criterion. We also give a best possible bound on the number of edges of an almost-planar graph.

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The Structure of Delta-Matroids with Width One Twists

The Electronic Journal of Combinatorics ◽

10.37236/7046 ◽

2018 ◽

Vol 25 (1) ◽

Author(s):

Carolyn Chun ◽

Rhiannon Hall ◽

Criel Merino ◽

Iain Moffatt ◽

Steven D Noble

Keyword(s):

Structure Theorem ◽

Feasible Set ◽

Excluded Minor ◽

The Difference

The width of a delta-matroid is the difference in size between a maximal and minimal feasible set. We give a Rough Structure Theorem for delta-matroids that admit a twist of width one. We apply this theorem to give an excluded minor characterisation of delta-matroids that admit a twist of width at most one.

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The excluded minor structure theorem with planarly embedded wall

Ars Mathematica Contemporanea ◽

10.26493/1855-3974.118.3dd ◽

2012 ◽

Vol 6 (2) ◽

pp. 187-196

Author(s):

Bojan Mohar

Keyword(s):

Structure Theorem ◽

Excluded Minor

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The Minor Crossing Number of Graphs with an Excluded Minor

The Electronic Journal of Combinatorics ◽

10.37236/728 ◽

2008 ◽

Vol 15 (1) ◽

Cited By ~ 1

Author(s):

Drago Bokal ◽

Gašper Fijavž ◽

David R. Wood

Keyword(s):

Crossing Number ◽

Excluded Minor ◽

A Minor

The minor crossing number of a graph $G$ is the minimum crossing number of a graph that contains $G$ as a minor. It is proved that for every graph $H$ there is a constant $c$, such that every graph $G$ with no $H$-minor has minor crossing number at most $c|V(G)|$.

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