  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 was an open problem for $P_k$-free graphs, $k \ge 5$. Recently,
Lokshtanov, Vatshelle, and Villanger proved that MWIS can be solved in
polynomial time for $P_5$-free graphs, and Lokshtanov, Pilipczuk, and van
Leeuwen proved that MWIS can be solved in quasi-polynomial time for $P_6$-free
graphs. It still remains an open problem whether MWIS can be solved in
polynomial time for $P_k$-free graphs, $k \geq 6$ or in quasi-polynomial time
for $P_k$-free graphs, $k \geq 7$. Some characterizations of $P_k$-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
($P_7$,triangle)-free graphs. This extends the corresponding result for
($P_6$,triangle)-free graphs and may provide some progress in the study of MWIS
for $P_7$-free graphs.
