literature it is assumed that a benzenoid graph can be embedded into the regular hexagonal lattice [@gucy-89]. Let $G$ be a benzenoid graph. Any inner face of $G$ is called a *hexagon* of $G$ and the number of all the hexagons of $G$ is denoted by $h(G)$. A vertex shared by three hexagons of $G$ is called an *internal* vertex of $G$. The number of all internal vertices of $G$ will be denoted by $n_i(G)$. A benzenoid graph is said to be *catacondensed* if it does not possess internal vertices. Otherwise it is called *pericondensed*.

We say that two faces of a plane graph are *adjacent* if they have at least one edge in common. The *dualist graph* of a given benzenoid graph $G$ consists of vertices corresponding to hexagons of $G$; two vertices are adjacent if and only if the corresponding hexagons are adjacent. Obviously, the dualist graph of $G$ is a tree if and only if $G$ is catacondensed. For a catacondensed benzenoid graph $G$, its dualistic tree has $h(G)$ vertices and none of its vertices have degree greater than three, see Figure \[benzenoid\].

![\[benzenoid\] A catacondensed benzenoid graph $G$ with its dualistic tree.](benzenoid.eps)

Let $G$ be a 