\[dodas\_enega\].*

    ![\[dodas\_enega\] A catacondensed benzenoid graph (or a phenylene) $G$ obtained from $G_0$ by Extension 2.](dodas_enega.eps)

For any $h \geq 2$, we define families ${\mathcal{B}}_h$ and ${\mathcal{P}}_h$ as follows.

-   If $h$ is an even number, then ${\mathcal{B}}_h$ (or ${\mathcal{P}}_h$) is the set of all the graphs that can be obtained from the catacondensed benzenoid graph with exactly two hexagons (or from the phenylene with exactly two hexagons) by performing Extension 1 exactly $\left(\frac{h}{2} - 1\right)$ times.

-   If $h$ is an odd number, then ${\mathcal{B}}_h$ (or ${\mathcal{P}}_h$) is the set of all the graphs that can be obtained from the catacondensed benzenoid graph with exactly two hexagons (or from the phenylene with exactly two hexagons) by performing Extension 1 exactly $\left(\frac{h-1}{2}-1\right)$ times and performing Extension 2 exactly once (in an arbitrary order).

For an example, Figure \[graf\_iz\_druzine\] shows a catacondensed benzenoid graph with nine hexagons that belongs to ${\mathcal{B}}_9$. Obviously, all the graphs in ${\mathcal{B}}_h \cup {\mathcal{P}}_h$, $h\geq 2$, have exactly $h$ hexagons.

![\[graf\_iz\_druzine\] A graph from ${\mathcal{B}}_9$.](graf_iz_druzine.eps)

The next lemma gives an upper bound on the number of segments.

\[st\_segmentov\] For a catacondensed benzenoid graph or a phenylene $G$ with $h(G) \geq 2$ it holds $s(G) \leq h(G)-1$. 