Department of Mathematics
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) ⩽ ⌈2n/3δ⌉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) ⩽ 2n/5, providing a new upper bound for cubic graphs.
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Independent domination number, Regular graph
Lam, Peter Che Bor, Wai Chee Shiu, and Liang Sun. "On independent domination number of regular graphs." Discrete Mathematics 202.1-3 (1999): 135-144.