Maximum Weight Independent Sets for (S_1,2,4,Triangle)-Free Graphs in Polynomial Time
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The Maximum Weight Independent Set (MWIS) problem on finite undirected graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum weight sum. MWIS is one of the most investigated and most important algorithmic graph problems; it is well known to be NP-complete, and it remains NP-complete even under various strong restrictions such as for triangle-free graphs. Its complexity for P_k-free graphs, k > 7, is an open problem. In 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 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 be solved in polynomial time for (S_1,2,3,bull)-free graphs. In this paper, using a similar approach as in BraMos2018, we show that MWIS can be solved in polynomial time for (S_1,2,4,triangle)-free graphs which generalizes the result for (P_7,triangle)-free graphs.
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