Polynomial-time algorithm for Maximum Independent Set in bounded-degree graphs with no long induced claws

TL;DR

This paper proves that the Maximum Independent Set problem can be solved in polynomial time for graphs with bounded degree that do not contain a long induced subdivided claw, advancing understanding of this complex problem.

Contribution

It establishes polynomial-time solvability of MWIS in bounded-degree, H-free graphs for any subdivided claw H, a significant step beyond previous quasipolynomial results.

Findings

MWIS is polynomial-time solvable in bounded-degree, H-free graphs for any subdivided claw H.

Extends known results from small paths to all subdivided claws in bounded-degree graphs.

Progress towards resolving MWIS complexity in H-free graphs with no long induced claws.

Abstract

For graphs $G$ and $H$, we say that $G$ is $H$-free if it does not contain $H$ as an induced subgraph. Already in the early 1980s Alekseev observed that if $H$ is connected, then the \textsc{Max Weight Independent Set} problem (MWIS) remains \textsc{NP}-hard in $H$-free graphs, unless $H$ is a path or a subdivided claw, i.e., a graph obtained from the three-leaf star by subdividing each edge some number of times (possibly zero). Since then determining the complexity of MWIS in these remaining cases is one of the most important problems in algorithmic graph theory. A general belief is that the problem is polynomial-time solvable, which is witnessed by algorithmic results for graphs excluding some small paths or subdivided claws. A more conclusive evidence was given by the recent breakthrough result by Gartland and Lokshtanov [FOCS 2020]: They proved that MWIS can be solved in quasipolynomial time in $H$-free graphs, where $H$ is any fixed path. If $H$ is an arbitrary subdivided claw, we know much less: The problem admits a QPTAS and a subexponential-time algorithm [Chudnovsky et al., SODA 2019]. In this paper we make an important step towards solving the problem by showing that for any subdivided claw $H$, MWIS is polynomial-time solvable in $H$-free graphs of bounded degree.