## 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 node 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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Pages (from-to) | 1182-1195 |

Number of pages | 14 |

Journal | Journal of Combinatorial Optimization |

Volume | 32 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2016 Nov 1 |

## Keywords

- Connectivity
- Diameter
- Dominating set
- Reconfiguration
- Reconfiguration graph
- Solution space