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Reconfiguration of Dominating Sets

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Computing and Combinatorics (COCOON 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8591))

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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.

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Aoba-yama 6-6-05, Aoba-ku, Sendai, 980-8579, Japan

    Akira Suzuki

  2. David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada

    Amer E. Mouawad & Naomi Nishimura

Authors
  1. Akira Suzuki
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  2. Amer E. Mouawad
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  3. Naomi Nishimura
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Georgia State University, 34 Peachtree Street, Suite 1410, 30303, Atlanta, GA, USA

    Zhipeng Cai

  2. Department of Computer Science, Georgia State University, 34 Peachtree Street, Suite 1443, 30303, Atlanta, GA, USA

    Alex Zelikovsky

  3. Department of Computer Science, Georgia State University, 34 Peachtree Street, Suite 1449, 30303, Atlanta, GA, USA

    Anu Bourgeois

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Suzuki, A., Mouawad, A.E., Nishimura, N. (2014). Reconfiguration of Dominating Sets. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds) Computing and Combinatorics. COCOON 2014. Lecture Notes in Computer Science, vol 8591. Springer, Cham. https://doi.org/10.1007/978-3-319-08783-2_35

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  • DOI: https://doi.org/10.1007/978-3-319-08783-2_35

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08782-5

  • Online ISBN: 978-3-319-08783-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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