Cluster Edge Deletion on 2-trees
The edge sets whose deletion leaves a cluster graph can easily be described as a formula with one free variable (the edge set) in monadic second-order graph logic. Therefore, by an optimization version of Courcelle's theorem, the minimum cardinality set can be found in linear time for graphs of bounded treewidth, which obviously include the 2-trees.

See e.g. Arnborg, Lagergren, and Seese, "Easy problems for tree-decomposable graphs", J. Algorithms 1991.