Maximum Weight Independent Sets for ($P_7$,Triangle)-Free Graphs in Polynomial Time

TL;DR

This paper proves that the Maximum Weight Independent Set problem can be solved in polynomial time for graphs that do not contain a path of length 7 or triangles, advancing understanding of MWIS complexity in restricted graph classes.

Contribution

The paper demonstrates polynomial-time solvability of MWIS for ($P_7$,triangle)-free graphs, extending previous results for ($P_6$,triangle)-free graphs and contributing to the open problem for $P_k$-free graphs.

Findings

MWIS is polynomial-time solvable for ($P_7$,triangle)-free graphs.

Extends previous results from ($P_6$,triangle)-free graphs.

Provides new insights into the complexity of MWIS in restricted graph classes.

Abstract

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.