Reconfiguration of dominating sets

Akira Suzuki, Amer E. Mouawad, Naomi Nishimura

Research output: Chapter in Book/Report/Conference proceedingConference contribution

8 Citations (Scopus)

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 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.

Original languageEnglish
Title of host publicationComputing and Combinatorics - 20th International Conference, COCOON 2014, Proceedings
PublisherSpringer-Verlag
Pages405-416
Number of pages12
ISBN (Print)9783319087825
DOIs
Publication statusPublished - 2014 Jan 1
Event20th International Computing and Combinatorics Conference, COCOON 2014 - Atlanta, GA, United States
Duration: 2014 Aug 42014 Aug 6

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8591 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference20th International Computing and Combinatorics Conference, COCOON 2014
CountryUnited States
CityAtlanta, GA
Period14/8/414/8/6

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Suzuki, A., Mouawad, A. E., & Nishimura, N. (2014). Reconfiguration of dominating sets. In 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