### The Inverse Domination in Semi-total Block Graphs

K. Ameenal Bibi, R. Selvakumar
2010 International Journal of Computer Applications
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 B 1 = {u 1 ,u 2 ,...,u r , r 2} be a block of G. Then we say that the point u 1 and block B 1 are incident with each other, as are u 2 and B 1, u 3 and B 1 and so on. If two distinct blocks B 1 and B 2 are incident with a common cut point then they are called adjacent blocks. Let B = {B 1 , B 2 ,...,B p } be the set of
more » ... s of G. The semi-total block graph T b (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 T b (G). A non-empty set D V B is a dominating set of T b (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 T b (G) is denoted by [T b (G)] and it is defined as the minimum cardinality taken over all the minimal dominating sets of T b (G). In this paper, we defined Inverse domination in semi-total block graphs. Let D be the minimum dominating set of T b (G). If (V B)-D contains a dominating set D' then D' is called the Inverse dominating set of T b (G). The Inverse domination number in semi-total block graph is denoted by '[T b (G)] and it is defined as the minimum cardinality taken over all the minimal Inverse dominating sets of T b (G). In this paper, many bounds on '[T b (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.