The diversity of domination Original Research Article

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 S 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 γH(G), γtH(G), and iH(G). We survey the applications and bounds obtained for the above domination parameters if H={Kn} or H = {Pi|2 ⩽ i ⩽ n}, for an integer n ⩾ 2.