Construction of a Maximum Stable Set with $k$-Extensions

  • Authors:

  • Peter L. Hammer;Igor E. Zverovich

  • Affiliations:

  • RUTCOR–Rutgers Center for Operations Research, Rutgers University, 640 Bartholomew Rd, Piscataway, NJ 08854-8003, USA (e-mail: hammer@rutcor.rutgers.edu and igor@rutcor.rutgers.edu);RUTCOR–Rutgers Center for Operations Research, Rutgers University, 640 Bartholomew Rd, Piscataway, NJ 08854-8003, USA (e-mail: hammer@rutcor.rutgers.edu and igor@rutcor.rutgers.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2005

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Abstract

A stable set $I$ of a graph $G$ is called $k$-extendable, $k \,{\ge}\, 1$, if there exists a stable set $X \,{\subseteq}\,V(G) {\setminus} I$ such that $|X| \,{\le}\, k$ and $|N(X) \,{\cap}\, I| \,{extendable if every stable set in $G$, which is not maximum, is $k$-extendable. Let us denote by ${\rm E}(k)$ the class of all $k$-extendable graphs.We present a finite forbidden induced subgraph characterization of the maximal hereditary subclass ${\rm PE}(k)$ in ${\rm E}(k)$ for every $k \,{\ge}\,1$.Thus, we define a hierarchy ${\rm PE}(1) \,{\subset}\, {\rm PE}(2) \,{\subset}\,{\cdots}\,{ \subset}\, {\rm PE}(k) \,{\subset}\,{ \cdots}\,$ of hereditary classes of graphs, in each of which a maximum stable set can be found in polynomial time. The hierarchy covers all graphs, and all its classes can be recognized in polynomial time.