Maximum Weight Independent Sets for (P7,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 triangle-free graphs. For a long time, its complexity was an open problem for Pk-free graphs, k≥5. Recently, Lokshtanov et al. (2014) proved that MWIS can be solved in polynomial time for P5-free graphs, Lokshtanov et al. (2015) proved that MWIS can be solved in quasi-polynomial time for P6-free graphs, and Bacsó et al. (2016), and independently, Brause (2017) extended this to Pk-free graphs for every fixed k. Then very recently, Grzesik et al. (2017) showed that MWIS can be solved in polynomial time for P6-free graphs. It still remains an open problem for Pk-free graphs, k≥7. 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 such as the recent result by Maffray and Pastor (2016) showing that MWIS can be solved in polynomial time for (P7,bull)-free graphs. © 2017 Elsevier B.V.