$G_2$ we have $|C_4(G_2)|=h(G_2)-1$, $|C_5(G_2)|=0$, $f(G_2) = 4h(G_2)-4$, and $|C_6(G_2)|=h(G_2)$. Also, it is easy to check that benzenoid graphs and phenylenes suffice the conditions of Theorem \[glavni\]. Therefore, the desired results follow by Theorem \[glavni\], Proposition \[gut-ben\], Proposition \[phen\], Lemma \[firzag1\], and Proposition \[second\_zagreb\].

To show one example, let $G$ be a benzenoid graph from Figure \[benzenoid\] and $G'$ a phenylene from Figure \[phenylene\]. As already noticed, $h(G)=h(G')=6$, $b(G)=b(G')=1$, and $s(G)=s(G')= 4$. Therefore, by Theorem \[Wie\_pol\] we compute $W_p(G) = 52$ and $W_p(G') = 72$.

In the rest of the section we will characterize the catacondensed benzenoid graphs and phenylenes with exactly $h$ hexagons for which the Wiener polarity index attains the minimum and the maximum value. Note that the catacondensed benzenoid graphs attaining the lower bound coincide with the graphs obtained in [@beh]. However, the result for the maximum value requires some additional insights.

If $G$ is a catacondensed benzenoid graph or a phenylene with exactly $h$ hexagons, we denote by $L_h$ the linear benzenoid chain or the linear phenylene chain, respectively, that also contains $h$ hexagons. Then $W_p(G) \geq W_p(L_h)$ and the equality holds if and only if $G \cong L_h$.

By Theorem \[Wie\_pol\], the Wiener polarity index 