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Keywords:

  • semi-topological r-wheel;
  • A-separator

Abstract

A semi-topological r-wheel, denoted by inline image, is a subdivision of the r-wheel preserving the spokes. This paper describes the r-connected graphs having no inline image-subgraphs. For inline image, these are shown to be only inline image, while the class inline image of 3-connected inline image-free graphs is unexpectedly rich. First, every graph G in inline image has an efficiently recognizable set of “contractible edges” (sometimes empty) such that a contraction minor inline image belongs to inline image if and only if F is a part of this set. So, the subclass inline image of ante-contraction members of inline image plays a key role. Second, the members of inline image have 3-edge cuts. The familiar cactus representation of minimum edge cuts (Dinits et al., in: Issledovaniya po Diskretnoy Optimizatsii, Nauka, Moscow, pp. 290–306, 1976 (Russian); also A. Schrijver, Combinatorial Optimization (Polyhedra and Efficiency), Algorithms and Combinatorics, Vol. 24, Springer, 2003, p. 253) maps inline image onto the class of trees whose internal vertices have even degrees, equal to 6 for any vertex adjacent to a leaf. The description of inline image (quite concise as expressed in appropriate terms) refers to the explicit reconstruction of the reverse image of such a tree. We also derive the upper bound inline image on the number of edges in an arbitrary n-vertex inline image-free graph, inline image, and conjecture that its maximum equals inline image.