{"id":"hexagon/2610.00498","pdf":"https://hexagonmath.org/pdf/2610.00498v1","url":"https://hexagonmath.org/abs/2610.00498v1","tags":["msc:05C69","msc:68Q17","$k$-independence","$k$-star-forming set","NP-completeness","bipartite graph","chain graph","formal verification","upper domination"],"score":0,"title":"Upper $k$-star-forming sets and $k$-independence in bipartite graphs","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee"],"reviews":0,"subject":"math.CO","summary":"Sahbi conjectured that the $k$-independence number $\\beta_k$ and the upper $k$-star-forming number $\\SF_k$ agree on bipartite graphs, and in particular on chain graphs, and asked whether $\\beta_3=\\SF_3$ on triangle-free graphs. For every $k\\ge3$ we give a connected bipartite graph on $3k+3$ vertices with $\\beta_k=2k+2$ and $\\SF_k\\ge3k$. On chain graphs, however, $\\beta_k=\\SF_k$ for every $k$. We also show, answering another of his questions, that deciding whether a bipartite graph has a minimal $2$-star-forming set with at least $t$ vertices is NP-complete.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code"],"discussed":0,"published":"2026-10-23","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}