The research of M.L. was partially supported by ISF grant 836/08.
On r-Connected Graphs with No Semi-topological r-Wheel
Article first published online: 6 AUG 2012
© 2012 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 72, Issue 3, pages 267–290, March 2013
How to Cite
Horev, E. and Lomonosov, M. (2013), On r-Connected Graphs with No Semi-topological r-Wheel. J. Graph Theory, 72: 267–290. doi: 10.1002/jgt.21642
- Issue published online: 23 JAN 2013
- Article first published online: 6 AUG 2012
- Manuscript Revised: 9 OCT 2011
- Manuscript Received: 11 NOV 2009
- Israel Scientific Foundation
- semi-topological r-wheel;
A semi-topological r-wheel, denoted by , is a subdivision of the r-wheel preserving the spokes. This paper describes the r-connected graphs having no -subgraphs. For , these are shown to be only , while the class of 3-connected -free graphs is unexpectedly rich. First, every graph G in has an efficiently recognizable set of “contractible edges” (sometimes empty) such that a contraction minor belongs to if and only if F is a part of this set. So, the subclass of ante-contraction members of plays a key role. Second, the members of 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 onto the class of trees whose internal vertices have even degrees, equal to 6 for any vertex adjacent to a leaf. The description of (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 on the number of edges in an arbitrary n-vertex -free graph, , and conjecture that its maximum equals .