Inverse Domination in X-Trees and Sibling Trees

A set D of vertices in a graph G is a dominating set if every vertex not in D is adjacent to at least one vertex in D. The minimum cardinality of a dominating set in G is called the domination number and is denoted by γ(G). Let D be a minimum dominating set of G. If $V-D$ contains a dominating set say D' of G, then D' is called an inverse dominating set with respect to D. The inverse domination number γ'(G) is the cardinality of a minimum inverse dominating set of G. A dominating set D is called a connected dominating set or an independent dominating set of G according as the induced subgraph D is connected or independent in G. The minimum of the cardinalities of the connected dominating sets of G or the independent dominating sets of G is called the connected domination number γc (G) or the independent domination number γᵢ (G) respectively. In this paper, we determine the inverse domination numbers in X-Trees and Sibling Trees. We have also determined the independent domination numbers of both the trees and the connected domination number of Sibling Trees. A result on inverse domination number of some classes of Hypertrees is also included.