Maximum Weight Independent Sets for ($S_{1,2,4}$,Triangle)-Free Graphs in Polynomial Time

TL;DR

This paper proves that the Maximum Weight Independent Set problem can be solved efficiently in polynomial time for a new class of graphs that exclude certain small subgraphs, extending previous results in graph theory.

Contribution

It introduces a polynomial-time algorithm for MWIS on ($S_{1,2,4}$,triangle)-free graphs, generalizing earlier results for ($P_7$,triangle)-free graphs.

Findings

MWIS solvable in polynomial time for ($S_{1,2,4}$,triangle)-free graphs

Extends previous polynomial-time results to a broader class of graphs

Provides new techniques for handling specific graph restrictions

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 for $P_k$-free graphs, $k \ge 7$, is an open problem. In \cite{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 \cite{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 \cite{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.