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 − 2δ for n/4 ≤ δ ≤ n/3, while i(G) ≤ δ for n/3 < δ ≤ 2n/5. Moreover, the extremal graphs achieving these upper bounds are also characterized. Copyright © 2010 Elsevier B.V. All rights reserved.
CitationWai Chee Shiu, Xue-gang Chen, Wai Hong Chan (2010). Triangle-free graphs with large independent domination number. Discrete Optimization, 7(1-2), 86-92. doi: 10.1016/j.disopt.2010.02.004
- Independent domination number
- Triangle-free graphs