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 Dk (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 DΓ(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 Dγ(G)+1(G) has exponential diameter, for γ(G) the minimum size of a dominating set. On the positive side, we show that Dn-μ (G) is connected and of linear diameter for any graph G on n vertices with a matching of size at least μ+1. © 2014 Springer International Publishing Switzerland.
CITATION STYLE
Suzuki, A., Mouawad, A. E., & Nishimura, N. (2014). Reconfiguration of dominating sets. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8591 LNCS, pp. 405–416). Springer Verlag. https://doi.org/10.1007/978-3-319-08783-2_35
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