Triangle-free graphs with large independent domination number

Abstract

Let G be a simple graph of order n and minimum degree δ . 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 δ , then i ( G ) ≤ n − 2 δ for n / 4 ≤ δ ≤ n / 3 , while i ( G ) ≤ δ for n / 3 < δ ≤ 2 n / 5 . Moreover, the extremal graphs achieving these upper bounds are also characterized.