  Let $K_d$ be the complete metric graph on $d$ vertices. We compute the
gonality of graphs obtained from $K_d$ by omitting edges forming a $K_h$, or
general configurations of at most $d-2$ edges. We also investigate if these
graphs can be lifted to curves with the same gonality. We lift the former
graphs and the ones obtained by removing up to $d-2$ edges not forming a $K_3$
using models of plane curves with certain singularities. We also study the
gonality when removing $d-1$ edges not forming a $K_3$. We use harmonic
morphism to lift these graphs to curves with the same gonality because in this
case plane singular models can no be longer used due to a result of Coppens and
Kato.
