TY - GEN

T1 - Reconfiguration of dominating sets

AU - Suzuki, Akira

AU - Mouawad, Amer E.

AU - Nishimura, Naomi

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-319-08783-2_35

DO - 10.1007/978-3-319-08783-2_35

M3 - Conference contribution

AN - SCOPUS:84904727189

SN - 9783319087825

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 405

EP - 416

BT - Computing and Combinatorics - 20th International Conference, COCOON 2014, Proceedings

PB - Springer-Verlag

T2 - 20th International Computing and Combinatorics Conference, COCOON 2014

Y2 - 4 August 2014 through 6 August 2014

ER -