Trees With Equal Domination and Tree-Free Domination Numbers
The tree-free domination number y(G; -Fk), k ≥ 2, of a graph G is the minimum cardinality of a dominating set S in G such that the subgraph (S) induced by S contains no tree on k vertices as a (not necessarily induced) subgraph (equivalently, each component of (S) has cardinality less than k). When k = 2, the tree-free domination number is the independent domination number. We obtain a characterization of trees with equal domination and tree-free domination numbers. This generalizes a result of Cockayne et al. (A characterisation of (y,i)-trees. J. Graph Theory 34(4) (2000) 277-292).
Haynes, Teresa W.; and Henning, Michael A.. 2002. Trees With Equal Domination and Tree-Free Domination Numbers. Discrete Mathematics. Vol.242(1-3). 93-102. https://doi.org/10.1016/S0012-365X(01)00053-X ISSN: 0012-365X