Maximum Weight Independent Sets for (, triangle)-free graphs in polynomial time

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 trianglefree graphs. Its complexity was an open problem for Pk-free graphs, k ≥ 5. Recently, Lokshtanov et al. [26] proved that MWIS can be solved in polynomial time for P5-free graphs, and Lokshtanov et al. [25] proved that MWIS can be solved in quasi-polynomial time for P6-free graphs. It still remains an open problem whether MWIS can be solved in polynomial time for Pk-free graphs, k ≥ 6 or in quasi-polynomial time for Pk-free graphs, k ≥ 7. Some characterizations of Pk-free graphs and some progress are known in the literature but so far did not solve the problem. In this paper, we show that MWIS can be solved in polynomial time for (P7,triangle)-free graphs. This extends the corresponding result for (P6,triangle)-free graphs and may provide some progress in the study of MWIS for P7-free graphs.