Restricted domination in graphs with minimum degree 2

The k-restricted domination number of a graph G   is the smallest integer dkdk such that given any subset U of k vertices of G, there exists a dominating set of G   of cardinality at most dkdk containing U. Hence, the k-restricted domination number of a graph G measures how many vertices are necessary to dominate a graph if an arbitrary set of k   vertices must be included in the dominating set. When k=0k=0, the k  -restricted domination number is the domination number. For k⩾1k⩾1, it is known that dk⩽(2n+3k)/5dk⩽(2n+3k)/5 for all connected graphs of order n and minimum degree at least 2 (see [M.A. Henning, Restricted domination in graphs, Discrete Math. 254 (2002) 175–189]). In this paper we characterize those graphs of order n   which are edge-minimal with respect to satisfying the conditions of connected, minimum degree at least two, and dk=(2n+3k)/5dk=(2n+3k)/5. These results extend results due to McCuaig and Shepherd [Domination in graphs with minimum degree two, J. Graph Theory 13 (1989) 749–762].