An ${\cal O}(n\sqrt{m})$ algorithm for the weighted stable set problem in {claw, net}-free graphs with $\alpha(G) \ge 4$

TL;DR

This paper introduces an efficient ${ m O}(n\sqrt{m})$ algorithm for solving the weighted stable set problem specifically in connected claw and net-free graphs with independence number at least 4, leveraging their structural properties.

Contribution

The paper characterizes the structure of certain claw and net-free graphs and develops a new algorithm exploiting this structure to efficiently solve the maximum weight stable set problem.

Findings

The algorithm runs in ${ m O}(n\sqrt{m})$ time.

Connected claw and net-free graphs with $ ext{alpha}(G) extgreater= 4$ can be decomposed into a strongly bisimplicial clique and at most two clique-strips.

Structural properties enable efficient optimization in these graph classes.

Abstract

In this paper we show that a connected {claw, net}-free graph $G(V, E)$ with $\alpha(G) \ge 4$ is the union of a strongly bisimplicial clique $Q$ and at most two clique-strips. A clique is strongly bisimplicial if its neighborhood is partitioned into two cliques which are mutually non-adjacent and a clique-strip is a sequence of cliques $\{H_0, \dots, H_p\}$ with the property that $H_i$ is adjacent only to $H_{i-1}$ and $H_{i+1}$. By exploiting such a structure we show how to solve the Maximum Weight Stable Set Problem in such a graph in time ${\cal O}(|V|\sqrt{|E|})$.