Abstract
The k-dominating graph \(D_k(G)\) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k, two such sets being adjacent if they differ by either adding or deleting a single vertex. A graph is a dominating graph if it is isomorphic to \(D_k(G)\) for some graph G and some positive integer k. Answering a question of Haas and Seyffarth for graphs without isolates, it is proved that if G is such a graph of order \(n\ge 2\) and with \(G\cong D_k(G)\), then \(k=2\) and \(G \cong K_{1,n-1}\) for some \(n\ge 4\). It is also proved that for a given r there exist only a finite number of r-regular, connected dominating graphs of connected graphs. In particular, \(C_6\) and \(C_8\) are the only dominating graphs in the class of cycles. Some results on the order of dominating graphs are also obtained.
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Acknowledgements
The authors would like to express their gratitude to the referees for their careful reading and helpful comments. S.K. acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0297) and that the project (Combinatorial Problems with an Emphasis on Games, N1-0043) was financially supported by the Slovenian Research Agency.
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Alikhani, S., Fatehi, D. & Klavžar, S. On the Structure of Dominating Graphs. Graphs and Combinatorics 33, 665–672 (2017). https://doi.org/10.1007/s00373-017-1792-5
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DOI: https://doi.org/10.1007/s00373-017-1792-5