Let G = (V, E) be a simple, finite, undirected graph with |V|= n and |E| = m. Kulli introduced the new graph valued function namely the semi-total block graph of a graph G. Let B1 = {u1,u2,...,ur, r ≥ 2} be a block of G. Then we say that the point u1 and block B1 are incident with each other, as are u2 and B1, u3 and B1 and so on. If two distinct blocks B1 and B2 are incident with a common cut point then they are called adjacent blocks. Let B = {B1, B2,...,Bp} be the set of blocks of G. The semi-total block graph Tb (G) of a graph G is the graph whose point set is V(G)  B(G) in which any two points are either adjacent or the corresponding members of G are incident. The points and blocks of G are members of Tb(G). A non-empty set DVB is a dominating set of Tb(G) if every point in (VB)-D is adjacent to atleast one point in D (Muddebihal, M.H. et al 2004). The domination number of Tb(G) is denoted by [Tb(G)] and it is defined as the minimum cardinality taken over all the minimal dominating sets of Tb(G). In this paper, we defined Inverse domination in semi-total block graphs. Let D be the minimum dominating set of Tb(G). If (VB)-D contains a dominating set D' then D' is called the Inverse dominating set of Tb(G). The Inverse domination number in semi-total block graph is denoted by '[Tb(G)] and it is defined as the minimum cardinality taken over all the minimal Inverse dominating sets of Tb(G). In this paper, many bounds on '[Tb(G)] are attained and its exact values for some standard graphs are found. Its relationships with other parameters are investigated. Nordhaus-Gaddum type results are also obtained for this parameter.

Domination number Inverse domination number semi-total block graph independence number