On the Forcing Domination and the Forcing Total Domination Numbers of a Graph

Let G be a connected graph with at least two vertices and S a gamma(t)-set of G. A subset T subset of S is called a forcing subset for S if S is the unique gamma(t)-set containing T. The forcing total domination number of S, denoted by f(gamma t)(S), is the cardinality of a minimum forcing subset of S. The forcing total domination number of G, denoted by f(gamma t)(G) is defined by f(gamma t)(G) = min {f(gamma t)(S)}, where the minimum is taken over all minimum total dominating sets S in G. Some general properties satisfied by this concepts are studied. The forcing total dominating number of certain standard graphs are determined. It is shown that for every pair a, b of integers with 0 <= a < b and b >= 1, there exists a connected graph G such that f(gamma t)(G) = a and gamma(t)(G) = b, where gamma(t)(G) is total domination number of G. It is also 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 t)(G)=a and f(gamma)(G) = b, where f(gamma)(G) is the forcing domination number of G.