THE FORCING NONSPLIT DOMINATION NUMBER OF A GRAPH

A dominating set S of a graph G is said to be nonsplit dominating set if the subgraph < V - S > is connected. The minimum cardinality of a nonsplit dominating set is called the nonsplit domination number and is denoted by gamma(ns) (G). For a minimum nonsplit dominating set S of G, a set T subset of S is called a forcing subset for S if S is the unique gamma(ns)-set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing nonsplit domination number of S, denoted by f(gamma ns )(S), is the cardinality of a minimum forcing subset of S. The forcing nonsplit domination number of G, denoted by f(gamma ns) (G) is defined by f(gamma ns) (G) = min{f(gamma ns) (S)} , where the minimum is taken over all gamma(ns)-sets S in G. The forcing nonsplit domination number of certain standard graphs are determined. It is shown that, for every pair of positive integers a and b with 0 <= a <= b and b >= 1, there exists a connected graph G such that f(gamma ns )(G) = a and gamma(ns) (G) = b. It is shown that, for every integer a >= 0, there exists a connected graph G with f(gamma) (G) = f(gamma ns) (G) = a, where f(gamma)(G) is the forcing domination number of the graph. Also, it is shown that, for every pair a, b of integers with a >= 0 and b >= 0 there exists a connected graph G such that f(gamma )(G) = a and f(gamma ns)(G) = b.