The diversity of domination

For any graph G and a set ℋ of graphs, two distinct vertices of G are said to be ℋ-adjacent if they are contained in a subgraph of G which is isomorphic to a member of ℋ. A set 5 of vertices of G is an ℋ-dominating set (total ℋ-dominating set) of G if every vertex in V(G) - S (V(G), respectively) is ℋ-adjacent to a vertex in S. An ℋ-dominating set of G in which no two vertices are ℋ-adjacent in G is an ℋ-independent dominating set of G. The minimum cardinality of an ℋ-dominating set, total ℋ-dominating set and ℋ-independent dominating set of G is known as the ℋ-domination number, total ℋ-domination number, and ℋ-independent dominating number, of G, denoted, respectively, by γ_ℋ(G), γ^tℋ(G), and i_ℋ(G). We survey the applications and bounds obtained for the above domination parameters if ℋ={K_n} or ℋ = {P_i | 2 ≤ i ≤ n}, for an integer n ≥ 2.