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A lower bound on the double outer-independent domination number of a tree

Krzykowski M.

Demonstratio Mathematica

2012

Vol. 45, nr 1

17-23

A vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D, and the set V (G) \ D is independent. The double outer-independent domination number of a graph G, denoted by (…) (G), is the minimum cardinality of a double outer-independent dominating set of G. We prove that for every nontrivial tree T of order n, with l leaves and s support vertices we have (…) (T) _ (2n+l .s +2)/3, and we characterize the trees attaining this lower bound. We also give a constructive characterization of trees T such that(…) (T) = (2n + 2)/3.