Information Processing Letters 86 (2003) 311–316 www.elsevier.com/locate/ipl Finding augmenting chains in extensions of claw-free graphs A. Hertz a,1 , V. Lozin b,2 , D. Schindl c,∗,3 a GERAD-École Polytechnique, 3000 chemin de la Cote Sainte-Catherine, Montréal, QC, Canada H3T 2A7 b RUTCOR, Rutgers University, Bartholomew Rd., Piscataway, NJ 08854-8003, USA c Institute of Mathematics, Swiss Federal Institute of Technology, CH-1015 Lausanne, Switzerland Received 2 September 2002; received in revised form 8 January 2003 Communicated by S. Albers Abstract Finding augmenting chains is in the heart of the maximum matching problem, which is equivalent to the maximum stable set problem in the class of line graphs. Due to the celebrated result of Edmonds, augmenting chains can be found in line graphs in polynomial time. Minty and Sbihi generalized this result to claw-free graphs. In this paper we extend it to larger classes. As a particular consequence, a new polynomially solvable case for the maximum stable set problem has been detected.  2003 Elsevier Science B.V. All rights reserved. Keywords: Stable sets; Augmenting chains; Polynomial algorithms; Combinatorial problems; Graph algorithms 1. Introduction We consider simple undirected graphs without loops and multiple edges. As usual, P n is the chordless chain (path) on n vertices. By S i,j,k we denote a tree with exactly three vertices of degree one being at distance i,j,k from the only vertex of degree three. In particular, S 1,1,1 is a claw, and S 1,1,2 is a fork. A banner is the graph with vertices a,b,c,d,e and * Corresponding author. E-mail address: david.schindl@epfl.ch (D. Schindl). 1 Research of the first author has been subsidized by NSERC Grant number 105384-02. 2 Research of the second one has been partially supported by the Office of Naval Research (Grant N00014-92-J-1375) and the National Science Foundation (Grant DMS-9806389). 3 Author acknowledges a partial support of the Swiss National Science Foundation, subsidy 2100-63409.00, holder D. de Werra, Swiss Federal Institute of Technology. edges (a,b), (b,c), (c,d), (d,e) and (e,b). By N(v) we denote the neighborhood of a vertex v, i.e., the subset of vertices adjacent to v. A matching in a graph is a subset of edges no two of which have a vertex in common, and a stable set is a subset of pairwise non-adjacent vertices. The problem of finding a matching of maximum cardinality is a special case of the maximum stable set problem, when restricted to the class of line graphs. In general, the maximum stable set problem is NP-hard, while the maximum matching problem is polynomially solvable. The first polynomial time algorithm to find a maximum matching has been proposed by Edmonds [5]. The algorithm exploits the idea of Berge that a matching M in a graph is maximum if and only if there are no augmenting (alternating) chains for M [3]. Later, this idea has been used independently by Minty [10] and Sbihi [16] in order to extend Edmonds’ result to a polynomial 0020-0190/03/$ – see front matter  2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0020-0190(03)00223-0