Independent Sets in Classes Related to Chair/Fork-free Graphs

TL;DR

This paper proves that the Maximum Weight Independent Set problem can be solved efficiently in certain graph classes that exclude specific subgraphs, extending previous results in graph theory and algorithm design.

Contribution

The paper establishes polynomial-time solvability of MWIS in ($S_{1, 2, 2}$, $S_{1, 1, 3}$, co-chair)-free graphs, broadening the classes of graphs with known efficient algorithms.

Findings

MWIS is polynomial-time solvable in the specified graph class.

Structural analysis of subclasses enables efficient algorithms.

Extends known results in graph class restrictions.

Abstract

The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. MWIS is known to be $NP$-complete in general, even under various restrictions. Let $S_{i,j,k}$ be the graph consisting of three induced paths of lengths $i, j, k$ with a common initial vertex. The complexity of the MWIS problem for $S_{1, 2, 2}$-free graphs, and for $S_{1, 1, 3}$-free graphs are open. In this paper, we show that the MWIS problem can solved in polynomial time for ($S_{1, 2, 2}$, $S_{1, 1, 3}$, co-chair)-free graphs, by analyzing the structure of the subclasses of this class of graphs. This extends some known results in the literature.