Article ID Journal Published Year Pages File Type
4949891 Discrete Applied Mathematics 2017 9 Pages PDF
Abstract

We show that interval graphs on n vertices have at most 3n/3≈1.4422n minimal dominating sets, and that these can be enumerated in time O∗(3n/3). As there are examples of interval graphs that actually have 3n/3 minimal dominating sets, our bound is tight. We show that the same upper bound holds also for trees, i.e. trees on n vertices have at most 3n/3≈1.4422n minimal dominating sets. The previous best upper bound on the number of minimal dominating sets in trees was 1.4656n, and there are trees that have 1.4167n minimal dominating sets. Hence our result narrows this gap. On general graphs there is a larger gap, with 1.7159n being the best known upper bound, whereas no graph with 1.5705n or more minimal dominating sets is known.

Related Topics
Physical Sciences and Engineering Computer Science Computational Theory and Mathematics
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