A fusene is a simple graph that is planar and biconnected, embedded in the plane with all vertices of vertex degree 2 or 3, all bounded faces hexagons (not necessarily regular polygons), and all vertices not in the boundary of the outer face of vertex degree 3 (Brinkmann et al. 2002).
Fusenes that are a subgraph of the regular hexagonal grid are called benzenoids.
Fusenes are perfect.
Let the number of internal vertices of a fusene be denoted 
. Then catafusenes (or catacondensed
fusenes) have 
(and are therefore also called "tree-like"), and perifusenes (or pericondensed
fusenes) have 
.
The numbers of catafusenes composed of 
hexagons are sometimes called
Harary-Read numbers, and have the generating function
![H(x)=1/(24)x^(-2){12+24x+48x^2-24x^3+[(1-x)(1-5x)]^(3/2)-3(5x+3)sqrt((1-x^2)(1-5x^2))-4sqrt((1-x^3)(1-5x^3))}-4
=x+x^2+2x^3+5x^4+12x^5+37x^6+...](/images/equations/Fusene/NumberedEquation1.svg) |
(OEIS A002216; Harary and Read 1970, Cyvin et al. 1993).
Polyhexes may also be classified on the basis of being geometrically planar (called nonhelicenic) or geometrically nonplanar (called helicenic). Fusenes include the helicenes. Here geometric planarity concerns the hexagon system, while every fusene is planar as an abstract graph.
The following table gives the numbers of fusenes with 
hexagons (Brinkmann et al.
2002, 2003), catafusenes (Harary and Read 1970, Beineke and Pippert 1974, Knop
et al. 1984, Cyvin et al. 1993), geometrically planar catafusenes,
and benzenoids.
 | fusenes | catafusenes | planar catafusenes | benzenoids |
| Sloane | A108070 | A002216 | A038142 | A018190 |
| 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 |
| 3 | 3 | 2 | 2 | 3 |
| 4 | 7 | 5 | 5 | 7 |
| 5 | 22 | 12 | 12 | 22 |
| 6 | 82 | 37 | 36 | 81 |
| 7 | 339 | 123 | 118 | 331 |
| 8 | 1505 | 446 | 411 | 1435 |
| 9 | 7036 | 1689 | 1489 | 6505 |
| 10 | 33836 | 6693 | 5572 | 30086 |
| 11 | 166246 | 27034 | 141229 | |
| 12 | 829987 | 111630 | 669584 | |
| 13 | 4197273 | 467262 | 3198256 | |
| 14 | 21456444 | 1981353 | 15367577 | |
| 15 | 110716585 | 8487400 | 74207910 | |
| 16 | 576027737 | 36695369 | 359863778 |
