A fusene is a simple planar 2-connected graph embedded in the plane with all vertices of degree 2 or 3, all bounded faces (not necessarily regular) hexagons, and all vertices not in the boundary of the outer face of degree 3 (Brinkmann et al. 2002).
Fusenes that are a subgraph of the regular hexagonal lattice are called benzenoids.
Fusenes are perfect.
Let the number of internal vertices of a polyhex 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 
polyhexes are sometimes called Harary-Read numbers, and have
the impressive 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.
The following table gives the numbers of 
-hexagon fusenes (Brinkmann et al. 2002, 2003) catafusenes
(Harary and Read 1970, Beinecke and Pippert 1974, Knop et al. 1984, Cyvin
et al. 1993), catafusenes, planar catafusenes, and simple catafusenes.
 | fusenes | catafusenes | planar catafusenes | simpl. catafusenes |
| 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 |
