Moreover, for all the graphs in ${\mathcal{B}}_h \cup{\mathcal{P}}_h$, $h \geq 2$, the equality holds.

If $G$ has exactly two hexagons, then $s(G)=1$ and the statement holds. Whenever we add exactly one hexagon to a graph, the number of segments stays the same or increases by one. Therefore, by induction we obtain $s(G) \leq h(G)-1$ for any $G$.

From the definition of the families ${\mathcal{B}}_h$ and ${\mathcal{P}}_h$ it follows that whenever we add two hexagons to a smaller graph, the number of segments increases by two, and whenever we add one hexagon, the number of segments increases by one. Therefore, the equality holds for these graphs.

Finally, everything is prepared for the following theorem.

If $G$ is a catacondensed benzenoid graph (or a phenylene) with $h$ hexagons, $h \geq 2$, and if $G'$ is an arbitrary graph from ${\mathcal{B}}_h$ (or ${\mathcal{P}}_h$), then $W_p(G) \leq W_p(G')$ and the equality holds if and only if $G \in {\mathcal{B}}_h$ (or $ G \in {\mathcal{P}}_h$).

By Theorem \[Wie\_pol\], the maximum value of the Wiener polarity index is attained by a graph $G$ which attains also the maximum value with respect to the number $s(G) + b(G)$.

By the construction, the graphs in ${\mathcal{B}}_h \cup{\mathcal{P}}_h$ obviously 