A characterization of trees with equal independent domination and secure domination numbers

Let i(G) and gamma(s)(G) be the independent domination number and secure domination number of a graph G, respectively. Merouane and Chellali (2015) [12] proved that i(T) <= gamma(s)(T) for any tree T and asked to characterize the trees T with i(T) = gamma(s)(T). In this paper, we answer the question. We introduce three operations on trees and prove that any tree T with i(T) = gamma(s)(T) can be obtained by these operations. (C) 2016 Elsevier B.V. All rights reserved.National Natural Science Foundation of China [61672050, 61472012, 61572046, 61502012]; National Basic Research Program (973 Program) of China [2013CB329600]; China Postdoctoral Science Foundation [2016M591013]SCI(E)ARTICLE14-1811