## Abstract

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted WCW(G). Let B be a complete bipartite induced subgraph of G on vertex sets of bipartition B_{X} and B_{Y}. Then B is generating if there exists an independent set S such that S∪ B_{X} and S∪ B_{Y} are both maximal independent sets of G. In the restricted case that a generating subgraph B is isomorphic to K_{1 , 1}, the unique edge in B is called a relating edge. Deciding whether an input graph G is well-covered is co-NP-complete. Therefore finding WCW(G) is co-NP-hard. Deciding whether an edge is relating is NP-complete. Therefore, deciding whether a subgraph is generating is NP-complete as well. In this article we discuss the connections among these problems, provide proofs for NP-completeness for several restricted cases, and present polynomial characterizations for some other cases.

Original language | English |
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Pages (from-to) | 2384-2399 |

Number of pages | 16 |

Journal | Algorithmica |

Volume | 80 |

Issue number | 8 |

DOIs | |

State | Published - 1 Aug 2018 |

## Keywords

- Generating subgraph
- Maximal independent set
- Relating edge
- Vector space
- Weighted well-covered graph