### Abstract

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of k, we consider properties of D_{k} (G), the graph consisting of a vertex for each dominating set of size at most k and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that D_{Γ(G)+1}(G) is not necessarily connected, for Γ(G) the maximum cardinality of a minimal dominating set in G. The result holds even when graphs are constrained to be planar, of bounded tree-width, or b-partite for b≥3. Moreover, we construct an infinite family of graphs such that D_{γ(G)+1}(G) has exponential diameter, for γ(G) the minimum size of a dominating set. On the positive side, we show that D_{n-μ} (G) is connected and of linear diameter for any graph G on n vertices with a matching of size at least μ+1.

Original language | English |
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Title of host publication | Computing and Combinatorics - 20th International Conference, COCOON 2014, Proceedings |

Publisher | Springer-Verlag |

Pages | 405-416 |

Number of pages | 12 |

ISBN (Print) | 9783319087825 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

Event | 20th International Computing and Combinatorics Conference, COCOON 2014 - Atlanta, GA, United States Duration: 2014 Aug 4 → 2014 Aug 6 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8591 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 20th International Computing and Combinatorics Conference, COCOON 2014 |
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Country | United States |

City | Atlanta, GA |

Period | 14/8/4 → 14/8/6 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Computing and Combinatorics - 20th International Conference, COCOON 2014, Proceedings*(pp. 405-416). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8591 LNCS). Springer-Verlag. https://doi.org/10.1007/978-3-319-08783-2_35