Abstract
A graph is well-dominated if all of its minimal dominating sets have the same cardinality. It is proved that there are exactly eleven connected, well-dominated, triangle-free graphs whose domination number is at most 3. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show that the Cartesian product of two connected, triangle-free graphs is well-dominated if and only if both graphs are complete graphs of order 2. Under the assumption that at least one of the connected graphs G or H has no isolatable vertices, we prove that the direct product of G and H is well-dominated if and only if either \(G=H=K_3\) or \(G=K_2\) and H is either the 4-cycle or the corona of a connected graph. Furthermore, we show that the disjunctive product of two connected graphs is well-dominated if and only if one of the factors is a complete graph and the other factor has domination number at most 2.
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Acknowledgements
The authors thank the anonymous referees for a careful reading of the paper. In particular, we thank one of the referees for suggesting a way to shorten and improve the proof of Theorem 8.
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Anderson, S.E., Kuenzel, K. & Rall, D.F. On Well-Dominated Graphs. Graphs and Combinatorics 37, 151–165 (2021). https://doi.org/10.1007/s00373-020-02235-z
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DOI: https://doi.org/10.1007/s00373-020-02235-z