Domination versus independent domination in regular graphs

A set S of vertices in a graph G is a dominating set if every vertex of G is in S or is adjacent to a vertex in S . If, in addition, S is an independent set, then S is an independent dominating set. The domination number γ ( G ) of G is the minimum cardinality of a dominating set in G , while the independent domination number i ( G ) of G is the minimum cardinality of an independent dominating set in G . We prove that for all integers k ≥ 3 it holds that if G is a connected k ‐regular graph, theni ( G )γ ( G )≤ k 2 , with equality if and only if G = K k , k . The result was previously known only for k ≤ 6 . This affirmatively answers a question of Babikir and Henning.