Abstract   Let  G  be a simple graph. The independent domination number  i ( G ) is the minimum cardinality among all maximal independent sets of  G . Haviland (1995) conjectured that any connected regular graph  G  of order  n  and degree  δ  ⩽  n /2 satisfies  i ( G ) ⩽ ⌈2 n /3 δ ⌉ δ /2. In this paper, we will settle the conjecture of Haviland in the negative by constructing counterexamples. Therefore a larger upper bound is expected. We will also show that a connected cubic graph  G  of order  n  ⩾ 8 satisfies  i ( G ) ⩽ 2 n /5, providing a new upper bound for cubic graphs.
['Peter Che Bor Lam', 'Wai Chee Shiu', 'Liang Sun']
On independent domination number of regular graphs