We define a 2-parametric hierarchy CLAP (m,n) of bi-hereditary classes of graphs, and show that a maximum stable set can be found in polynomial time within each class CLAP (m,n) . The classes can be recognized in polynomial time. | Yugoslav Journal of Operations Research 14 (2004), Number 1, 27-32 BI-INDUCED SUBGRAPHS AND STABILITY NUMBER* I. E. ZVEROVICH, O. I. ZVEROVICH RUTCOR – Rutgers Center for Operations Research, Rutgers University, Piscataway, New Jersey, USA igor@ Received: June 2003 / Accepted: February 2004 Abstract: We define a 2-parametric hierarchy CLAP (m, n) of bi-hereditary classes of graphs, and show that a maximum stable set can be found in polynomial time within each class CLAP (m, n) . The classes can be recognized in polynomial time. Keywords: Stability number, hereditary class, bi-hereditary class, forbidden induced subgraphs, forbidden bi-induced subgraphs. 1. INTRODUCTION A set S ⊆ V (G ) in a graph G is stable (or independent) if S does not contain adjacent vertices. A stable set of a graph G is called maximal if it is not contained in another stable set of G. A stable set of a graph G is called maximum if G does not have a stable set containing more vertices. The cardinality of a maximum stable set in G is the stability number of G, and it is denoted by α (G ) . Decision Problem 1 (Stable Set). Instance: A graph G and an integer k. Question: Is there a stable set in G with at least k vertices? This problem is known to be NP-complete (Karp [7], see also Garey and Johnson [3]). A class P of graphs is α-polynomial if there exists a polynomial-time algorithm to solve Stable Set Problem within P. We shall define a hierarchy CLAP (m, n) of α-polynomial graph classes. The hierarchy covers all graphs. * The first author was supported by DIMACS Winter 2002/2003 Award. AMS Subject Classification: 05C69. 28 . Zverovich, . Zverovich / Bi-Induced Subgraphs and Stability Number Note that it is easy to find the stability number of graphs in any class without large connected induced bipartite subgraphs. In other words, the class CONNBIP ( N ) free graphs is α-polynomial, where CONNBIP ( N ) is the set of all connected bipartite graphs of order N. Lozin
