  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 complexity of the MWIS problem for $P_6$-free graphs is unknown. In
this note, we show that the MWIS problem can be solved in time $O(n^3m)$ for
($P_6$, banner)-free graphs by analyzing the structure of subclasses of these
class of graphs. This extends the existing results for ($P_5$, banner)-free
graphs, and ($P_6$, $C_4$)-free graphs. Here, $P_t$ 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.
