Maximum Weight Independent Sets in ( S_1, 1, 3 , bull)-free Graphs

The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. The MWIS problem is well known to be NP-complete in general, even under substantial restrictions. The computational complexity of the MWIS problem for \(S_{1, 1, 3}\)-free graphs is unknown. In this note, we give a proof for the solvability of the MWIS problem for (\(S_{1, 1, 3}\), bull)-free graphs in polynomial time. Here, an \(S_{1, 1, 3}\) is the graph with vertices \(v_1, v_2, v_3, v_4, v_5, v_6\) and edges \(v_1v_2, v_2v_3, v_3v_4, v_4v_5, v_4v_6\), and the bull is the graph with vertices \(v_1, v_2, v_3, v_4, v_5\) and edges \(v_1v_2, \) \( v_2v_3, v_3v_4, \) \( v_2v_5, v_3v_5\).