Infeasibility of degree-4 Steiner points in optimal BCST for α in (0.5, 1) on the plane

Prove or disprove that, in the branched central spanning tree (BCST) problem with terminals embedded in the Euclidean plane and parameter α in the open interval (0.5, 1), every optimal solution uses Steiner points of degree at most three unless a Steiner point coincides with a terminal (i.e., establish the infeasibility of degree-4 Steiner points under these conditions).

Background

The paper proves that degree-4 Steiner points cannot occur in optimal BCST solutions on the Euclidean plane when α ∈ [0, 0.5] or α = 1, unless they collapse with a terminal. For the intermediate regime α ∈ (0.5, 1), the authors provide empirical evidence but no analytical proof.

Resolving this would close the theoretical gap for the entire range α ∈ [0,1] and align the structural characterization of optimal solutions with the established results for the endpoint cases.