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Journal of Combinatorial Optimization

, Volume 32, Issue 4, pp 1182–1195 | Cite as

Reconfiguration of dominating sets

  • Akira Suzuki
  • Amer E. Mouawad
  • Naomi Nishimura
Article

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 \(D_{\varGamma (G)+1}(G)\) is not necessarily connected, for \(\varGamma (G)\) 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 \(b \ge 3\). Moreover, we construct an infinite family of graphs such that \(D_{\gamma (G)+1}(G)\) has exponential diameter, for \(\gamma (G)\) the minimum size of a dominating set. On the positive side, we show that \(D_{n-\mu }(G)\) is connected and of linear diameter for any graph G on n vertices with a matching of size at least \(\mu +1\).

Keywords

Dominating set Reconfiguration Reconfiguration graph Connectivity Diameter Solution space 

Notes

Acknowledgments

Research supported by JSPS Grant-in-Aid for Scientific Research, Grant Number 26730001, and the Natural Science and Engineering Research Council of Canada.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Akira Suzuki
    • 1
    • 2
  • Amer E. Mouawad
    • 3
  • Naomi Nishimura
    • 3
  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
  2. 2.CREST, JSTSaitamaJapan
  3. 3.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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