The structure and labelled enumeration of K_{3,3}-subdivision-free projective-planar graphs

Structure and labelled enumeration of K3,3-subdivision-free projective-planar graphs

We consider the class F of 2-connected non-planar K3,3-subdivision-free graphs that are embeddable in the projective plane. We show that these graphs admit a unique decomposition as a graph K5 (the core) where the edges are replaced by two-pole networks constructed from 2-connected planar graphs. A method to enumerate these graphs in the labelled case is described. Moreover, we enumerate the ho...

Counting Labelled Projective-planar Graphs without a K 3,3 -subdivision *

We consider the class F of labelled 2-connected non-planar graphs without a K3,3-subdivision that are embeddable in the projective plane. A method of enumerating these graphs is described. We also enumerate the homeomorphically irreducible graphs in F and homeomorphically irreducible 2-connected planar graphs. The methods are based on the projective-planarity characterization for graphs without...

The structure of K3, 3-subdivision-free toroidal graphs

We describe the structure of 2-connected non-planar toroidal graphs with no K3,3-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 K5-subdivision in [A. Gagarin and W. Kocay, “Embedding graphs containing K5-subdivisions”...

Exponential Speedup of Fixed Parameter Algorithms on K_{3,3}-minor-free or K_5-minor-free Graphs

$n$-Array Jacobson graphs

We generalize the notion of Jacobson graphs into $n$-array columns called $n$-array Jacobson graphs and determine their connectivities and diameters. Also, we will study forbidden structures of these graphs and determine when an $n$-array Jacobson graph is planar, outer planar, projective, perfect or domination perfect.