Outer-k-connected component domination in graphs

A subset S of the vertices of a graph G is an outer-connected dominating set, if S is a dominating set of G and G − S is connected. The outer-connected domination number of G, denoted by γ̃c(G), is the minimum cardinality of an OCDS of G. In this paper we generalize the outer-connected domination in graphs. Many of the known results and bounds of outer-connected domination number are immediate consequences of our results. Let k ≥ 1 be an integer. A subset D of vertices of G is an outer-k-connected component dominating set if D is a dominating and the graph G − D has exactly k connected component. The outer-k-connected component domination number of G, denoted by γ̃ c (G), is the minimum cardinality of a outer-k-connected component dominating set of G. We study the outer-kconnected component domination in a graph G. We present properties and bounds of outerk-connected component domination number in graphs, and show that the decision problem for the outer-k-connected component domination number of an arbitrary graph G is NPcomplete. Finally, we determine γ̃ c (G) for several certain classes of graphs G.