Upper k-star-forming sets and k-independence in bipartite graphs

Sahbi conjectured that the k-independence number and the upper k-star-forming number agree on bipartite graphs, and in particular on chain graphs, and asked whether they agree for k = 3 on triangle-free graphs. For every k >= 3 we give a connected bipartite graph on 3k + 3 vertices with k-independence number 2k + 2 and a minimal k-star-forming set of size 3k. On chain graphs, however, the two numbers agree for every k. Finally, deciding whether a bipartite graph has a minimal 2-star-forming set of a given size is NP-complete. The results are formalised in Lean 4 with Mathlib, the complexity result through the identity it rests on.