have the maximum number of branched hexagons among all the catacondensed benzenoid graphs or phenylenes with $h$ hexagons. If $h$ is even, these graphs are the only graphs attaining the upper bound with respect to the number of branched hexagons. If $h$ is odd, the upper bound with respect to the number of branched hexagons is attained also by some other graphs. To describe them, we first introduce Extension 3.

**Extension 3:** *Let $G_0$ be a catacondensed benzenoid graph (or a phenylene) and let $h_0$ be an arbitrary terminal hexagon of $G_0$. We denote by $G$ a graph obtained from $G_0$ by attaching exactly one new hexagon to $h_0$ such that $h_0$ becomes a linear hexagon (and in the case of phenylenes, we also add a quadrilateral between $h_0$ and the new hexagon). See Figure \[dodas\_ravno\].*

![\[dodas\_ravno\] A catacondensed benzenoid graph (or a phenylene) $G$ obtained from $G_0$ by Extension 3.](dodas_ravno.eps)

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 3 