Gonality of curves whose normalizations are one or two copies of ${\mathbb{P}}^1$

We study the gonality of curves $C$ over $\mathbb{C}$ whose normalization is composed of one or two copies of ${\mathbb{P}}^1$. In the first case, $C$ is a nodal curve with $g(C)$ nodes, and in the second case $C$ is a so-called binary curve. In any case we show that the usual bound $\mathrm{g}\mathrm{o}\mathrm{n}(C)\le \lfloor \frac{g(C)+3}{2}\rfloor$ holds if $g(C)\ge 2$, with equality holding generically.