Outer-2-independent domination in graphs

We initiate the study of outer-2-independent domination in graphs. An outer-2-independent dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)?D has a neighbor in D and the maximum vertex degree of the subgraph induced by V(G)?D is at most one. The outer-2-independent domination number of a graph G is the minimum cardinality of an outer-2-independent dominating set of G. We show that if a graph has minimum degree at least two, then its outer-2-independent domination number equals the number of vertices minus the 2-independence number. Then we investigate the outer-2-independent domination in graphs with minimum degree one. We also prove the Vizing-type conjecture for outer-2-independent domination and disprove the Vizing-type conjecture for outer-connected domination.