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 $Hsubseteq G$ 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).
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