Graph and Order Theory ‘model one of Topology’s simpler ends’. It then makes sense to consider properties of graphs that are Topologically invariant. This standardly begins with graph homeomorphs, and the corresponding homeomorph irreducibles (HI): a type of ‘prime’ . Moreover, since specialized types of graph or furtherly structured variants of graphs are useful in many modelling situations, we should work out in detail analogues of this structural study for each specialization or variant of note.

Simple graphs

Chapter 44 of Combinatorics | for homeomorphs and homeomorph irreducibles (HI).

Chapter 60 of Combinatorics | for foliations.

OEAGOT Graph Double Irreducibility Classes | including foliation irreducibles (FI) and double irreducibles (DI): concurrently HI and FI.

OEAGOT Graph Smallest Double Irreducible Arenas | on N = 0 to 5 vertices.

Basic variants

OEAGOT Rooted and Di Graphs Double Irreducibility Classes |

Girtheomorphs (girth-preserving) and pohomeomorphs (of posets!)

Cubic graphs

Many advanced studies in Graph Theory are for simple graphs that are furthermore cubic.

– Cubeomorphs and cubic irreducibles (CI).

– And remodelling graphs beyond cubic in terms of cubic graphs!