953 research outputs found
Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs
We describe the structure of 2-connected non-planar toroidal graphs with no
K_{3,3}-subdivisions, using an appropriate substitution of planar networks into
the edges of certain graphs called toroidal cores. The structural result is
based on a refinement of the algorithmic results for graphs containing a fixed
K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing
K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these
graphs in linear-time and makes possible to enumerate labelled 2-connected
toroidal graphs containing no K_{3,3}-subdivisions and having minimum vertex
degree two or three by using an approach similar to [A. Gagarin, G. Labelle,
and P. Leroux, "Counting labelled projective-planar graphs without a
K_{3,3}-subdivision", submitted, arXiv:math.CO/0406140, (2004)].Comment: 18 pages, 7 figures and 4 table
The obstructions for toroidal graphs with no 's
Forbidden minors and subdivisions for toroidal graphs are numerous. We
consider the toroidal graphs with no -subdivisions that coincide with
the toroidal graphs with no -minors. These graphs admit a unique
decomposition into planar components and have short lists of obstructions. We
provide the complete lists of four forbidden minors and eleven forbidden
subdivisions for the toroidal graphs with no 's and prove that the
lists are sufficient.Comment: 10 pages, 7 figures, revised version with additional detail
Bipartite Minors
We introduce a notion of bipartite minors and prove a bipartite analog of
Wagner's theorem: a bipartite graph is planar if and only if it does not
contain as a bipartite minor. Similarly, we provide a forbidden minor
characterization for outerplanar graphs and forests. We then establish a
recursive characterization of bipartite -Laman graphs --- a certain
family of graphs that contains all maximal bipartite planar graphs.Comment: 9 page
Counting unlabelled toroidal graphs with no K33-subdivisions
We provide a description of unlabelled enumeration techniques, with complete
proofs, for graphs that can be canonically obtained by substituting 2-pole
networks for the edges of core graphs. Using structure theorems for toroidal
and projective-planar graphs containing no K33-subdivisions, we apply these
techniques to obtain their unlabelled enumeration.Comment: 25 pages (some corrections), 4 figures (one figure added), 3 table
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