Polynomial-time solvability of Triangle Partition at one edge edit

Determine whether Sidestep(Triangle Partition, dist, 1) belongs to FP, where Triangle Partition asks whether an n-vertex graph with n divisible by three can be partitioned into triangles and dist is the maximum-degree distance between graphs.

Background

Triangle Partition is an NP-hard problem in which the vertices of a graph must be partitioned into triangles. The paper observes that radius 2 is polynomial-time tractable under the maximum-degree distance. It asks whether tractability already holds at radius 1, leaving this threshold unresolved.