Weighted independent sets in a subclass of P6-free graphs

TheMaximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. The complexity of the MWIS problem for P6-free graphs is unknown. In this note, we show that the MWIS problem can be solved in time O(nm) for (P6, banner)-free graphs by analyzing the structure of subclasses of these class of graphs. This extends the existing results for (P5, banner)-free graphs, and (P6, C4)-free graphs. Here, Pt denotes the chordless path on t vertices, and a banner is the graph obtained from a chordless cycle on four vertices by adding a vertex that has exactly one neighbor on the cycle.