  Catacondensed benzenoids (those benzenoids having no carbon atom belonging to
three hexagonal rings) form the simplest class of polycyclic aromatic
hydrocarbons (PAH). They have a long history of study and are of wide chemical
importance. In this paper, mathematical possibilities for natural extension of
the notion of a catacondensed benzenoid are discussed, leading under plausible
chemically and physically motivated restrictions to the notion of a
catacondensed chemical hexagonal complex (CCHC). A general polygonal complex is
a topological structure composed of polygons that are glued together along
certain edges. A polygonal complex is flat if none of its edges belong to more
than two polygons. A connected flat polygonal complex determines an orientable
or nonorientable surface, possibly with boundary. A CCHC is then a connected
flat polygonal complex all of whose polygons are hexagons and each of whose
vertices belongs to at most two hexagonal faces. We prove that all CCHC are
Kekulean and give formulas for counting the perfect matchings in a series of
examples based on expansion of cubic graphs in which the edges are replaced by
linear polyacenes of equal length. As a preliminary assessment of the likely
stability of molecules with CCHC structure, all-electron quantum chemical
calculations are applied to molecular structures based on several CCHC, using
either linear or kinked unbranched catafused polyacenes as the expansion motif.
The systems examined all have closed shells according to H\"uckel theory and
all correspond to minima on the potential surface, thus passing the most basic
test for plausibility as a chemical species.
