### Abstract

A dominating setD of a graph G is a subset of V (G) such that for every vertex v ∈ V (G), either v ∈ D or there exists a vertex u ∈ D that is adjacent to v in G. Dominating sets of small cardinality are of interest. A connected dominating setC of a graph G is a dominating set of G such that the subgraph induced by the vertices of C in G is connected. A weakly-connected dominating setW of a graph G is a dominating set of G such that the subgraph consisting of V (G) and all edges incident with vertices in W is connected. In this paper we present several algorithms for finding small connected dominating sets and small weakly-connected dominating sets of regular graphs. We analyse the average-case performance of these heuristics on random regular graphs using differential equations, thus giving upper bounds on the size of a smallest connected dominating set and the size of a smallest weakly-connected dominating set of random regular graphs.

Original language | English |
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Pages (from-to) | 2305-2322 |

Number of pages | 18 |

Journal | Discrete Mathematics |

Volume | 309 |

Issue number | 8 |

DOIs | |

Publication status | Published - 28 Apr 2009 |

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## Cite this

*Discrete Mathematics*,

*309*(8), 2305-2322. https://doi.org/10.1016/j.disc.2008.05.029