The maximum Kendall-Tau distance between two permutations allowed by an acyclic constraint graph equals the number of incomparable pairs exactly when the induced poset has dimension at most 2.
Harris, Erik Insko, and Darleen Perez-Lavin,Peak sets of classical Coxeter groups, Involve 10 (2017), no
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Metrics on Permutation Families Defined by a Restriction Graph
The maximum Kendall-Tau distance between two permutations allowed by an acyclic constraint graph equals the number of incomparable pairs exactly when the induced poset has dimension at most 2.