2-Edge connected dominating sets and 2-Connected dominating sets of a graph

A (k)-connected (resp. (k)-edge connected) dominating set (D) of a connected graph (G) is a subset of (V(G)) such that (G[D]) is (k)-connected (resp. (k)-edge connected) and each (vin V(G)backslash D) has at least one neighbor in (D). The (k) -connected domination number (resp. (k) -edge connected domination number) of a graph (G) is the minimum size of a (k)-connected (resp. (k)-edge connected) dominating set of (G), and denoted by (gamma _k(G)) (resp. (gamma '_k(G))). In this paper, we investigate the relation of independence number and 2-connected (resp. 2-edge-connected) domination number, and prove that for a graph (G), if it is (2)-edge connected, then (gamma '_2(G)le 4alpha (G)-1), and it is (2)-connected, then (gamma _2(G)le 6alpha (G)-3), where (alpha (G)) is the independent number of (G).