TY - GEN
T1 - Reconfiguration of dominating sets
AU - Suzuki, Akira
AU - Mouawad, Amer E.
AU - Nishimura, Naomi
PY - 2014
Y1 - 2014
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 -