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The Maximum Weight Independent Set (MWIS) problem on finite undirected graphswith vertex weights asks for a set of pairwise nonadjacent vertices of maximumweight sum. MWIS is one of the most investigated and most important algorithmicgraph problems; it is well known to be NP-complete, and it remains NP-completeeven under various strong restrictions such as for triangle-free graphs. Itscomplexity for $P_k$-free graphs, $k \ge 7$, is an open problem. In\cite{BraMos2018}, it is shown that MWIS can be solved in polynomial time for($P_7$,triangle)-free graphs. This result is extended by Maffray and Pastor\cite{MafPas2016} showing that MWIS can be solved in polynomial time for($P_7$,bull)-free graphs. In the same paper, they also showed that MWIS can besolved in polynomial time for ($S_{1,2,3}$,bull)-free graphs.  In this paper, using a similar approach as in \cite{BraMos2018}, we show thatMWIS can be solved in polynomial time for ($S_{1,2,4}$,triangle)-free graphswhich generalizes the result for ($P_7$,triangle)-free graphs.

Maximum Weight Independent Sets for ($S_{1,2,4}$,Triangle)-Free Graphs in Polynomial Time