Triangle-free graphs with large independent domination number

Let GG be a simple graph of order nn and minimum degree δδ. The independent domination number i(G)i(G) is defined as the minimum cardinality of an independent dominating set of GG. We prove the following conjecture due to Haviland [J. Haviland, Independent domination in triangle-free graphs, Discrete Mathematics 308 (2008), 3545–3550]: If GG is a triangle-free graph of order nn and minimum degree δδ, then i(G)≤n−2δi(G)≤n−2δ for n/4≤δ≤n/3n/4≤δ≤n/3, while i(G)≤δi(G)≤δ for n/3<δ≤2n/5n/3<δ≤2n/5. Moreover, the extremal graphs achieving these upper bounds are also characterized.