The domination number of a graph (Formula presented.), denoted (Formula presented.), is the minimum size of a dominating set of (Formula presented.), and the independent domination number of (Formula presented.), denoted (Formula presented.), is the minimum size of a dominating set of (Formula presented.) that is also independent. Let (Formula presented.) be an integer. Generalizing a result on cubic graphs by Lam, Shiu, and Sun, we prove that (Formula presented.) for a connected (Formula presented.) -regular graph (Formula presented.) that is not (Formula presented.), which is tight for (Formula presented.). This answers a question by Goddard et al. in the affirmative. We also show that (Formula presented.) for a connected (Formula presented.) -regular graph (Formula presented.) that is not (Formula presented.), strengthening upon a result of Knor, Škrekovski, and Tepeh. In addition, we prove that a graph (Formula presented.) with maximum degree at most 4 satisfies (Formula presented.), which is also tight.
Eun\u2010Kyung Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF\u20102020R1I1A1A0105858711). Ilkyoo Choi was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF\u20102018R1D1A1B07043049), and also by the Hankuk University of Foreign Studies Research Fund. Boram Park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF\u20102018R1C1B6003577).
