Maximum Weight Independent Set in lClaw-Free Graphs in Polynomial Time

Abstract

The Maximum Weight Independent Set (MWIS) problem is a well-known NP-hard problem. For graphs $G_1, G_2$, $G_1+G_2$ denotes the disjoint union of $G_1$ and $G_2$, and for a constant $l \ge 2$, $lG$ denotes the disjoint union of $l$ copies of $G$. A {\em claw} has vertices $a,b,c,d$, and edges $ab,ac,ad$. MWIS can be solved for claw-free graphs in polynomial time; the first two polynomial time algorithms were introduced in 1980 by \cite{Minty1980,Sbihi1980}, then revisited by \cite{NakTam2001}, and recently improved by \cite{FaeOriSta2011,FaeOriSta2014}, and by \cite{NobSas2011,NobSas2015} with the best known time bound in \cite{NobSas2015}. Furthermore MWIS can be solved for the following extensions of claw-free graphs in polynomial time: fork-free graphs \cite{LozMil2008}, $K_2$+claw-free graphs \cite{LozMos2005}, and apple-free graphs \cite{BraLozMos2010,BraKleLozMos2008}. This manuscript shows that for any constant $l$, MWIS can be solved for $l$claw-free graphs in polynomial time. Our approach is based on Farber's approach showing that every $2K_2$-free graph has ${\cal O}(n^2)$ maximal independent sets \cite{Farbe1989}, which directly leads to a polynomial time algorithm for MWIS on $2K_2$-free graphs by dynamic programming. Solving MWIS for $l$claw-free graphs in polynomial time extends known results for claw-free graphs, for $lK_2$-free graphs for any constant $l$ \cite{Aleks1991,FarHujTuz1993,Prisn1995,TsuIdeAriShi1977}, for $K_2$+claw-free graphs, for $2P_3$-free graphs \cite{LozMos2012}, and solves the open questions for $2K_2+P_3$-free graphs and for $P_3$+claw-free graphs being two of the minimal graph classes, defined by forbidding one induced subgraph, for which the complexity of MWIS was an open problem.