Triangle-free graphs with large independent domination number

Let G be a simple graph of order n and minimum degree delta. The independent domination number i(G) is defined as the minimum cardinality of an independent dominating set of G. We prove the following conjecture due to Haviland [J. Haviland, Independent domination in triangle-free graphs, Discrete Mathematics 308 (2008), 3545-3550]: If G is a triangle-free graph of order n and minimum degree delta, then i(G) <= n - 2 delta for n/4 <= delta <= n/3, while i(G) <= delta for n/3 < delta <= 2n/5. Moreover, the extremal graphs achieving these upper bounds are also characterized. (C) 2010 Elsevier B.V. All rights reserved.