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

A semi-topological r-wheel, denoted by Sr, is a subdivision of the r-wheel preserving the spokes; the paper describes the r-connected graphs having no Sr-subgraphs. For r > 3, these are shown to be only Kr,r, while the class H of 3-connected S3-free graphs is unexpectedly rich. First, every graph G in H has an efficiently recognizable set of “contractible edges” (sometimes empty) such that a contraction minor G/F belongs to H if and only if F is a part of this set. So, the subclass H of ante-contraction members of H plays a key role. Second, the members of H have 3-edge cuts. The familiar cactus representation of minimum edge cuts (E. Dinits et al.. In: Issledovaniya po Diskretnoy Optimizatsii (A. A. Friedman, ed.), “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 H onto the class of trees whose internal vertices have even degrees, equal to 6 for any vertex adjacent to a leaf. The description of H (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 (2r − 3)(n− r + 1) on the number of edges in an arbitrary n-vertex Sr-free graph, r ≥ 4, and conjecture that its maximum equals (r−1)(n−r+1)+ ⌊ r−1 2 ⌋