  By considering graphs as discrete analogues of Riemann surfaces, Baker and
Norine (Adv. Math. 2007) developed a concept of linear systems of divisors for
graphs. Building on this idea, a concept of gonality for graphs has been
defined and has generated much recent interest. We show that there are
connected graphs of treewidth 2 of arbitrarily high gonality. We also show that
there exist pairs of connected graphs $\{G,H\}$ such that $H\subseteq G$ and
$H$ has strictly lower gonality than $G$. These results resolve three open
problems posed in a recent survey by Norine (Surveys in Combinatorics 2015).
