Numerical semigroups with embedding dimension three and minimal catenary degree

We characterize numerical semigroups $S$ with embedding dimension three attaining equality in the inequality $\max\Delta(S)+2\leq \operatorname{cat}(S)$,where$\Delta(S)$denotestheDeltasetof$S$and$\operatorname{cat}(S)$denotesthecatenarydegreeof$S$.