## Abstract

Let G be a graph. A set S of vertices in G dominates the graph if every vertex of G is either in S or a neighbor of a vertex in S. Finding a minimum cardinality set which dominates the graph is an NP-complete problem. The graph G is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of recognizing well-dominated graphs is not known. We show that recognizing well-dominated graphs can be done polynomially for graphs without cycles of lengths 4 and 5, by proving that a graph belonging to this family is well-dominated if and only if it is well-covered. Assume that a weight function w is defined on the vertices of G. Then G is w-well-dominated if all its minimal dominating sets are of the same weight. We prove that the set of weight functions w such that G is w-well-dominated is a vector space, and denote that vector space by WWD(G). We show that WWD(G) is a subspace of WCW(G), the vector space of weight functions w such that G is w-well-covered. We provide a polynomial characterization of WWD(G) for the case that G does not contain cycles of lengths 4, 5, and 6.

Original language | English |
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Pages (from-to) | 1793-1801 |

Number of pages | 9 |

Journal | Discrete Mathematics |

Volume | 340 |

Issue number | 8 |

DOIs | |

State | Published - 1 Aug 2017 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2017 Elsevier B.V.

## Keywords

- Maximal independent set
- Minimal dominating set
- Vector space
- Well-covered graph
- Well-dominated graph