Some connectivity properties for excluded minors of the graph invariant v(G)

  • Authors:

  • Hein van der Holst

  • Affiliations:

  • Univerzita Karlova v Praze, Matematicko-fyzikální fakulta, Katedra Aplikované Matematiky, Malostranské nám. 25, 118 00 Praha 1, Czech Republic

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2003

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Abstract

Let v(G) be the graph invariant introduced by Colin de Verdière in J. Combin. Theory Ser. B. 74 (1998) 121. Let k ≥ 0, and let H be an excluded minor of the class of graphs G with v(G) ≤ k. We show that H has no vertex cuts of size at most two and that, if S is a vertex cut of size three of H, then G - S has two components, and S is the neighbourhood of a vertex v and the subgraph induced by S ∪ {v} is isomorphic to one of the graphs in a certain collection of six graphs.