An O(n2logn) algorithm for the weighted stable set problem in claw-free graphs

• Published in: Mathematical programming Vol. 186; no. 1-2; pp. 409 - 437
• Main Authors: Nobili, Paolo, Sassano, Antonio
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2021; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Decomposition; Full Length Paper; Graphs; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Stability; Theoretical; Matching; 05C85 Graph algorithms; cliques; Claw-free graphs; Quasi-line graphs; 05C69 Dominating sets; covering and packing; 05C70 Factorization; Stable set; independent sets
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-019-01461-5

A graph G ( V ,  E ) is claw-free if no vertex has three pairwise non-adjacent neighbours. The maximum weight stable set (MWSS) problem in a claw-free graph is a natural generalization of the matching problem and has been shown to be polynomially solvable by Minty and Sbihi in 1980. In a remarkable paper, Faenza, Oriolo and Stauffer have shown that, in a two-step procedure, a claw-free graph can be first turned into a quasi-line graph by removing strips containing all the irregular nodes and then decomposed into { claw , net }- free strips and strips with stability number at most three. Through this decomposition, the MWSS problem can be solved in O ( | V | ( | V | log | V | + | E | ) ) time. In this paper, we describe a direct decomposition of a claw-free graph into { claw , net }- free strips and strips with stability number at most three which can be performed in O ( | V | 2 ) time. In two companion papers we showed that the MWSS problem can be solved in O ( | E | log | V | ) time in claw-free graphs with α ( G ) ≤ 3 and in O ( | V | | E | ) time in {claw, net}-free graphs with α ( G ) ≥ 4 . These results prove that the MWSS problem in a claw-free graph can be solved in O ( | V | 2 log | V | ) time, the same complexity of the best and long standing algorithm for the MWSS problem in line graphs .