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On r-Connected Graphs with No Semi-topological r-Wheel


  • The research of M.L. was partially supported by ISF grant 836/08.


A semi-topological r-wheel, denoted by math formula, is a subdivision of the r-wheel preserving the spokes. This paper describes the r-connected graphs having no math formula-subgraphs. For math formula, these are shown to be only math formula, while the class math formula of 3-connected math formula-free graphs is unexpectedly rich. First, every graph G in math formula has an efficiently recognizable set of “contractible edges” (sometimes empty) such that a contraction minor math formula belongs to math formula if and only if F is a part of this set. So, the subclass math formula of ante-contraction members of math formula plays a key role. Second, the members of math formula 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 math formula onto the class of trees whose internal vertices have even degrees, equal to 6 for any vertex adjacent to a leaf. The description of math formula (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 math formula on the number of edges in an arbitrary n-vertex math formula-free graph, math formula, and conjecture that its maximum equals math formula.